🧠:[<think>Okay, let's see. The problem is asking which of the given inequalities is not true when we know that x is greater than y (x > y). The options are A, B, C, and D. I need to check each one and see if they hold true under the condition x > y. The one that doesn't hold true is the answer.Starting with option A: x - 6 > y - 6. Hmm, if we subtract the same number from both sides of an inequality, the inequality should still hold. Since x > y, subtracting 6 from both x and y shouldn't change the direction of the inequality. Let me test with numbers to be sure. Suppose x is 10 and y is 5. Then 10 - 6 = 4 and 5 - 6 = -1. So 4 > -1 is true. Another example: x = 3, y = 2. Then 3 - 6 = -3 and 2 - 6 = -4. -3 > -4 is still true. So A seems to be true. So A is not the answer.Moving to option B: 3x > 3y. If we multiply both sides of an inequality by a positive number, the direction stays the same. Since 3 is positive, multiplying x > y by 3 gives 3x > 3y. Let's check with numbers. Take x = 5, y = 2. Then 3*5=15, 3*2=6. 15 > 6, which is true. Another example: x = -1, y = -3 (since x > y even with negatives). 3*(-1)=-3 and 3*(-3)=-9. -3 > -9 is true. So B is also true. Not the answer.Option C: -2x < -2y. Here, we are multiplying both sides by a negative number (-2). When you multiply or divide an inequality by a negative number, the direction of the inequality flips. Since x > y, multiplying both sides by -2 would reverse it: -2x < -2y. Let's verify with numbers. Let x = 4, y = 1. Then -2*4 = -8 and -2*1 = -2. So -8 < -2, which is true. Another example: x = -2, y = -5 (since -2 > -5). Multiplying by -2: -2*(-2)=4 and -2*(-5)=10. So 4 < 10, which is true. So C is correct. So C is not the answer.Now option D: -3x + 6 > -3y + 6. Let's analyze this. Let's try to manipulate the inequality step by step. Starting with the original x > y.First, multiply both sides by -3. Remember, multiplying by a negative number reverses the inequality. So:x > y Multiply both sides by -3: -3x < -3yThen, add 6 to both sides: -3x + 6 < -3y + 6But the option D says -3x + 6 > -3y + 6, which is the opposite of what we derived. Therefore, D is not true.Wait, let me verify with numbers. Let's take x = 5 and y = 3 (since 5 > 3). Compute left side: -3*5 +6 = -15 +6 = -9. Right side: -3*3 +6 = -9 +6 = -3. So, -9 > -3? No, that's false. So D is indeed not true.Another example: x = 2, y = 1. Left side: -3*2 +6 = -6 +6 = 0. Right side: -3*1 +6 = -3 +6 = 3. So, 0 > 3? No, that's false. So D is not true.So the answer should be D.Final Answerboxed{D}</think>Given ( x > y ), we need to determine which of the given inequalities is not true.Option A: ( x - 6 > y - 6 )Subtracting 6 from both sides of ( x > y ) preserves the inequality:[ x - 6 > y - 6 ]This is true. Testing with values confirms this. For example, ( x = 10 ) and ( y = 5 ) gives ( 4 > -1 ), which is true.Option B: ( 3x > 3y )Multiplying both sides of ( x > y ) by 3 (a positive number) preserves the inequality:[ 3x > 3y ]This is true. Testing with values confirms this. For example, ( x = 5 ) and ( y = 2 ) gives ( 15 > 6 ), which is true.Option C: ( -2x < -2y )Multiplying both sides of ( x > y ) by -2 (a negative number) reverses the inequality:[ -2x < -2y ]This is true. Testing with values confirms this. For example, ( x = 4 ) and ( y = 1 ) gives ( -8 < -2 ), which is true.Option D: ( -3x + 6 > -3y + 6 )Starting with ( x > y ), multiplying by -3 reverses the inequality:[ -3x < -3y ]Adding 6 to both sides:[ -3x + 6 < -3y + 6 ]However, option D states ( -3x + 6 > -3y + 6 ), which is the opposite. Testing with values confirms this. For example, ( x = 5 ) and ( y = 3 ) gives ( -9 > -3 ), which is false.Thus, the inequality that is not true is (boxed{D}).,<think>Okay, let's see. The problem says: Given that x is greater than y, which of the following inequalities is not true? The options are A, B, C, D. Hmm. I need to check each one to see if they must be true given that x > y, and find the one that isn't necessarily true. Let me take them one by one.Starting with option A: x - 6 > y - 6. Well, if x > y, then if we subtract the same number from both sides, the inequality should still hold, right? Like, if you have two numbers and one is bigger, subtracting 6 from both shouldn't change their order. Let me test with numbers. Suppose x is 10 and y is 5. Then 10 - 6 = 4 and 5 - 6 = -1. So 4 > -1, which is true. What if x and y are negative? Let's say x is -1 and y is -2 (since x > y). Then -1 - 6 = -7, and -2 -6 = -8. So -7 > -8, which is also true. Seems like A is always true. So A is not the answer.Moving to option B: 3x > 3y. If we multiply both sides of an inequality by a positive number, the inequality direction stays the same. Since 3 is positive, multiplying x > y by 3 would give 3x > 3y, so this should be true. Testing with numbers: x=2, y=1. 3*2=6, 3*1=3. 6>3, yes. If x and y are negative, say x=-1, y=-2. 3*(-1)=-3, 3*(-2)=-6. -3 > -6, which is still true. So B is true. Not the answer.Option C: -2x < -2y. Here, we are multiplying both sides of the original inequality x > y by a negative number, -2. When you multiply or divide an inequality by a negative number, you have to reverse the inequality sign. So starting with x > y, multiply both sides by -2: -2x < -2y. So this should be correct. Let me verify with numbers. Take x=3, y=2. Then -2*3 = -6 and -2*2 = -4. So -6 < -4, which is true. If x and y are negative, like x=-1, y=-2. Then -2*(-1)=2, -2*(-2)=4. So 2 < 4? Yes, that's true. So C is also a true inequality. So C is not the answer.Now option D: -3x + 6 > -3y + 6. Let's see. Let me try to manipulate the inequality step by step. Starting with x > y. Let's see if we can derive D from x > y. If we can, then it's true; if not, maybe it's not.First, maybe we can subtract 6 from both sides of D? Let's see. The original D is -3x +6 > -3y +6. If we subtract 6 from both sides, we get -3x > -3y. Then, if we divide both sides by -3, we have to reverse the inequality sign, so x < y. But wait, our original assumption is x > y, which contradicts x < y. Therefore, this suggests that if we start with x > y, the inequality -3x +6 > -3y +6 would lead to x < y, which is false. Therefore, D is not true.But let me double-check with actual numbers to be sure. Let's pick some numbers where x > y. Let's take x=5 and y=3. Then compute D: -3(5) +6 = -15 +6 = -9, and -3(3)+6 = -9 +6 = -3. So -9 > -3? No, that's false. So in this case, D is not true. Another example: x=2, y=1. Then D: -3*2 +6 = -6 +6 = 0, and -3*1 +6 = -3 +6 = 3. So 0 > 3? No, false. How about if x and y are negative? Let x=-1, y=-2. Then D: -3*(-1) +6 = 3 +6 =9, and -3*(-2)+6=6 +6=12. So 9 >12? No, 9 is not greater than 12. So in both cases, D is not true. Therefore, D is the inequality that is not true when x > y.Wait, but let me check if there's a case where D might be true. Suppose x and y are such that when we apply the operations, maybe the numbers work out. Let me think. Suppose x is very close to y, like x= y + ε, where ε is a small positive number. Then, substituting into D: -3(y + ε) +6 > -3y +6. Simplify left side: -3y -3ε +6. Right side: -3y +6. So left side is -3y -3ε +6, right side is -3y +6. Therefore, subtracting right side from left side: -3ε, which is negative. Therefore, left side is less than right side, so inequality is false. So even if x is just a tiny bit bigger than y, D is still false. So D is never true when x > y. Therefore, the answer must be D.Final Answerboxed{D}</think>Given ( x > y ), we need to determine which inequality is not true.Option A: ( x - 6 > y - 6 ) Subtracting 6 from both sides of ( x > y ) preserves the inequality. Testing with values confirms this is true. Conclusion: True.Option B: ( 3x > 3y ) Multiplying both sides of ( x > y ) by 3 (a positive number) preserves the inequality. Testing with values confirms this is true. Conclusion: True.Option C: ( -2x < -2y ) Multiplying both sides of ( x > y ) by -2 (a negative number) reverses the inequality. Testing with values confirms this is true. Conclusion: True.Option D: ( -3x + 6 > -3y + 6 ) Subtracting 6 from both sides gives ( -3x > -3y ). Dividing by -3 reverses the inequality, leading to ( x < y ), which contradicts ( x > y ). Testing with values (e.g., ( x = 5 ), ( y = 3 )) shows the inequality is false. Conclusion: Not true.Thus, the inequality that is not true is (boxed{D}).]

🔑:Given x > y, we need to determine which of the following inequalities is not true.A: Starting with x > y, we subtract 6 from both sides to maintain the inequality, which gives us:[x - 6 > y - 6]Thus, option A is a true statement based on the given condition.B: Multiplying both sides of x > y by 3 (a positive number), we get:[3x > 3y]Therefore, option B is also a true statement based on the given condition.C: Multiplying both sides of x > y by -2 (a negative number) reverses the inequality, leading to:[-2x y and multiplying both sides by -3 (a negative number) reverses the inequality, so we get:[-3x y is:[boxed{D}]

🧠:[<think>Alright, let me try to solve this problem step by step. The question is asking for the value of the expression:[frac{0.00001 times (0.01)^2 times 1000}{0.001}]And the options are given as (a) through (e). Hmm, okay. Let me break this down.First, dealing with decimals can be a bit tricky, so maybe converting them into scientific notation would make it easier. Let me recall that scientific notation represents numbers as ( a times 10^n ), where ( 1 leq |a| < 10 ) and n is an integer. So let's convert each number in the expression to scientific notation.Starting with 0.00001. That's the same as 1 divided by 100,000, which is ( 1 times 10^{-5} ). Got that.Next, (0.01)^2. Well, 0.01 is ( 1 times 10^{-2} ). Squaring that would be ( (10^{-2})^2 = 10^{-4} ), so the square is ( 1 times 10^{-4} ).Then there's 1000. That's straightforward, it's ( 1 times 10^{3} ).In the denominator, we have 0.001, which is ( 1 times 10^{-3} ).So substituting all these into the original expression, we get:[frac{(10^{-5}) times (10^{-4}) times (10^{3})}{10^{-3}}]Now, let me simplify the numerator first. When multiplying exponents with the same base, we add the exponents. So:( 10^{-5} times 10^{-4} times 10^{3} = 10^{-5 + (-4) + 3} = 10^{-6} ).So the numerator simplifies to ( 10^{-6} ), and the denominator is ( 10^{-3} ).Therefore, the entire expression becomes:[frac{10^{-6}}{10^{-3}} = 10^{-6 - (-3)} = 10^{-3}]Wait, hold on. Dividing exponents: when you divide, you subtract the denominator's exponent from the numerator's. So ( 10^{-6} div 10^{-3} = 10^{-6 - (-3)} = 10^{-3} ).So that simplifies to ( 10^{-3} ), which is option (c). Hmm, let me verify this again step by step to make sure I didn't make a mistake.Original numbers converted to exponents:- 0.00001 = ( 10^{-5} )- (0.01)^2 = ( (10^{-2})^2 = 10^{-4} )- 1000 = ( 10^{3} )- Denominator: 0.001 = ( 10^{-3} )Multiplying numerator: ( 10^{-5} times 10^{-4} = 10^{-9} ), then ( 10^{-9} times 10^{3} = 10^{-6} ). Then dividing by denominator: ( 10^{-6} div 10^{-3} = 10^{-3} ). Yes, that seems right.Alternatively, let's compute without converting to exponents, just using decimals to check:0.00001 is 0.00001Multiply by (0.01)^2: 0.01 * 0.01 = 0.0001. So 0.00001 * 0.0001 = 0.0000000001Wait, hold on. Wait: 0.00001 is 1e-5, 0.0001 is 1e-4. So 1e-5 * 1e-4 = 1e-9. Then multiply by 1000 (1e3): 1e-9 * 1e3 = 1e-6. Then divide by 0.001 (1e-3): 1e-6 / 1e-3 = 1e-3. So same result. So 0.001, which is 1e-3. So the answer is 10^{-3}, which is option (c). Hmm.Wait, but let me do the decimal multiplication step by step to confirm.First, 0.00001 is 1e-5. Then (0.01)^2 is 0.0001. So 1e-5 * 0.0001. Let's compute that:0.00001 * 0.0001. Let's count the decimal places. 0.00001 has 5 decimal places, 0.0001 has 4 decimal places. Multiplying them, the total decimal places are 5 + 4 = 9. So 1e-5 * 1e-4 = 1e-9, which is 0.000000001. Then multiplying by 1000: 0.000000001 * 1000 = 0.000001. Then divide by 0.001: 0.000001 / 0.001. Dividing 0.000001 by 0.001. Let's convert them to fractions:0.000001 is 1e-6, and 0.001 is 1e-3. So 1e-6 / 1e-3 = 1e-3, which is 0.001. So 0.001 is 10^{-3}, so the answer is (c). Hmm, seems consistent.Alternatively, if I do the decimal division directly: 0.000001 divided by 0.001. Let's express both in terms of thousandths. 0.001 is 1 thousandth. 0.000001 is 1 millionth. So how many thousandths are there in a millionth? Since a millionth is 0.001 divided by 1000, so 0.000001 is 0.001 / 1000, so dividing that by 0.001 gives 1/1000, which is 0.001. So that's 10^{-3}. Yeah, that works.So regardless of the approach, it seems like the answer is 10^{-3}, which is option (c). Therefore, I think (c) is the correct answer.But let me double-check once more to be absolutely sure. Sometimes when dealing with exponents, signs can be confusing.Original expression:Numerator: 0.00001 * (0.01)^2 * 1000Denominator: 0.001Convert each to exponents:0.00001 = 1e-50.01 = 1e-2, squared is 1e-41000 = 1e3Multiply numerator: 1e-5 * 1e-4 = 1e-9; then 1e-9 * 1e3 = 1e-6Denominator: 0.001 = 1e-3Divide numerator by denominator: 1e-6 / 1e-3 = 1e-3. Yep, that's 10^{-3}.Alternatively, in decimal form:0.00001 is 0.00001Multiply by (0.01)^2: 0.00001 * 0.0001 = 0.000000001Multiply by 1000: 0.000000001 * 1000 = 0.000001Divide by 0.001: 0.000001 / 0.001 = 0.0010.001 is 10^{-3}, so the same result.Therefore, I can be confident that the answer is option (c), 10^{-3}.Final Answerboxed{C}</think>To solve the given problem, we start by converting each number into scientific notation:- (0.00001 = 10^{-5})- ((0.01)^2 = (10^{-2})^2 = 10^{-4})- (1000 = 10^3)- (0.001 = 10^{-3})Substituting these into the expression, we get:[frac{10^{-5} times 10^{-4} times 10^3}{10^{-3}}]Simplifying the numerator by adding the exponents:[10^{-5} times 10^{-4} times 10^3 = 10^{-5 + (-4) + 3} = 10^{-6}]Now, dividing by the denominator:[frac{10^{-6}}{10^{-3}} = 10^{-6 - (-3)} = 10^{-3}]Thus, the value of the expression is (10^{-3}), which corresponds to option (c).[boxed{C}],<think>Okay, so I need to figure out the value of this fraction: (0.00001 × (0.01)² × 1000) divided by 0.001. Hmm, let's break it down step by step. First, I should probably convert all these decimals into powers of 10 to make it easier. Let me recall that 0.1 is 10⁻¹, 0.01 is 10⁻², and so on.Starting with the numerator: 0.00001. Let's see, 0.00001 is the same as 1 divided by 100,000, which is 10⁵. So that would be 10⁻⁵. Got that. Then, there's (0.01)². Since 0.01 is 10⁻², squaring that would be (10⁻²)² = 10⁻⁴. Okay, next up is 1000. That's straightforward, 1000 is 10³. So multiplying those three parts in the numerator: 10⁻⁵ × 10⁻⁴ × 10³. When multiplying exponents with the same base, we add the exponents. So -5 + (-4) + 3 = -6. So the numerator simplifies to 10⁻⁶.Now the denominator is 0.001. Let me convert that to a power of 10 as well. 0.001 is 1/1000, which is 10⁻³. So now the entire expression is (10⁻⁶) divided by (10⁻³). When dividing exponents, we subtract them: -6 - (-3) = -3. So that would be 10⁻³. Therefore, the value of the expression is 10⁻³, which corresponds to option (c). Wait a second, let me double-check my steps to make sure I didn't make any mistakes. Starting with the numerator: 0.00001 is indeed 10⁻⁵. Then (0.01)²: 0.01 is 10⁻², squared is 10⁻⁴. 1000 is 10³. Multiplying those: 10⁻⁵ × 10⁻⁴ × 10³. Let me verify the exponent addition: -5 + (-4) + 3 = (-5 -4) +3 = -9 +3 = -6. So numerator is 10⁻⁶. Denominator is 0.001 = 10⁻³. So 10⁻⁶ / 10⁻³ = 10^(-6 +3) = 10⁻³. Yep, that seems right. Alternatively, maybe I can check using the original numbers without converting to exponents. Let's see:Numerator: 0.00001 × (0.01)² × 1000.First, calculate (0.01)². 0.01 × 0.01 = 0.0001. Then multiply by 0.00001: 0.00001 × 0.0001 = 0.0000000001. Then multiply by 1000: 0.0000000001 × 1000 = 0.0000001. So numerator is 0.0000001. Denominator is 0.001. So dividing 0.0000001 by 0.001. That's the same as 0.0000001 ÷ 0.001. Moving the decimal points: 0.0000001 ÷ 0.001 = 0.0001. Hmm, wait, 0.0001 is 10⁻⁴. Wait, that contradicts my previous result. Hmm, so now I have a problem. Which one is correct?Hold on, let me recalculate the decimal way. Let's go step by step.First, numerator:1. Start with 0.00001. That's 0.00001.2. Then multiply by (0.01)². 0.01 squared is 0.0001. So 0.00001 × 0.0001.Multiplying these: 0.00001 has 5 decimal places, 0.0001 has 4 decimal places. 1 × 1 = 1, and total decimal places is 5 + 4 = 9. So 0.000000001. Wait, 0.00001 × 0.0001 = 0.000000001? Let me check: 0.00001 is 1e-5, 0.0001 is 1e-4. 1e-5 * 1e-4 = 1e-9, which is 0.000000001. Yes, that's correct.Then multiply by 1000. So 0.000000001 × 1000 = 0.000001. So numerator is 0.000001.Denominator is 0.001. So 0.000001 divided by 0.001. Let's do this division.0.000001 ÷ 0.001. Let's convert both to fractions:0.000001 is 1/1,000,000 and 0.001 is 1/1,000. So dividing them: (1/1,000,000) ÷ (1/1,000) = (1/1,000,000) × (1,000/1) = 1/1,000. Which is 0.001. Which is 10⁻³. So that's 10⁻³. Wait, but when I did the decimal way step by step, I first thought numerator was 0.0000001, but actually, it's 0.000001. So maybe I miscalculated earlier.Wait, first time when I did it step by step with decimals, I messed up. Let me do it again carefully:Numerator: 0.00001 × (0.01)^2 × 1000.First, (0.01)^2 = 0.0001.Then, 0.00001 × 0.0001 = 0.0000000001? Wait, no. Let me multiply 0.00001 and 0.0001.0.00001 is 1e-5, 0.0001 is 1e-4. Multiplying them gives 1e-9, which is 0.000000001.Then, 0.000000001 × 1000 = 0.000001.Therefore, numerator is 0.000001, denominator is 0.001. So 0.000001 / 0.001 = 0.001. Which is 10⁻³. So that's the same as before.Wait, so my initial exponent method was correct. But when I tried to do the decimal multiplication the first time, I made a mistake. So the correct answer is 10⁻³, which is option (c). Therefore, my initial answer was correct. The mistake was in my second check where I incorrectly multiplied 0.00001 and 0.0001 as 0.0000000001 instead of 0.000000001. So that's where the confusion came from. So confirming now, both methods give 10⁻³, so answer is (c).But just to make sure, let's write out all the steps again clearly:Numerator:0.00001 × (0.01)² × 1000Convert each to scientific notation:0.00001 = 1 × 10⁻⁵(0.01)² = (1 × 10⁻²)² = 1 × 10⁻⁴1000 = 1 × 10³Multiply these together:10⁻⁵ × 10⁻⁴ × 10³ = 10^{(-5 -4 +3)} = 10^{-6}Denominator:0.001 = 1 × 10⁻³Divide numerator by denominator:10^{-6} / 10^{-3} = 10^{(-6 - (-3))} = 10^{-3}Thus, the answer is 10⁻³, which is option (c). So I think that's solid. The confusion came from a miscalculation when converting back to decimals, but upon careful rechecking, both methods align. So the correct answer is (c).Final Answerboxed{C}</think>To determine the value of (frac{0.00001 times (0.01)^{2} times 1000}{0.001}), we convert each number to powers of 10 and simplify step by step.1. Convert each term to scientific notation: - (0.00001 = 10^{-5}) - ((0.01)^2 = (10^{-2})^2 = 10^{-4}) - (1000 = 10^3) - (0.001 = 10^{-3})2. Simplify the numerator: [ 10^{-5} times 10^{-4} times 10^3 = 10^{-5 + (-4) + 3} = 10^{-6} ]3. Divide by the denominator: [ frac{10^{-6}}{10^{-3}} = 10^{-6 - (-3)} = 10^{-3} ]Thus, the value of the expression is (10^{-3}), which corresponds to option (c).[boxed{C}]]

🔑:We have:begin{aligned}frac{0.00001 times(0.01)^{2} times 1000}{0.001} & =frac{10^{-5} timesleft(10^{-2}right)^{2} times 10^{3}}{10^{-3}}=frac{10^{-5} times 10^{-4} times 10^{3}}{10^{-3}}= & =frac{10^{-5+(-4)+3}}{10^{-3}}=frac{10^{-6}}{10^{-3}}=10^{-6-(-3)}=10^{-3}end{aligned}The correct option is (c).

🧠:[<think>Okay, let's see. I need to figure out which of these statements A, B, C, D is true. Let me go through each one step by step.Starting with option A: The arithmetic square root of √4 is 2. Hmm. Wait, the arithmetic square root typically refers to the non-negative root. So √4 is already the arithmetic square root, right? And √4 equals 2. So statement A is saying the arithmetic square root of √4 is 2. That seems redundant. Wait, maybe there's a misunderstanding here. Let me clarify. The arithmetic square root of a number is the non-negative root. So the arithmetic square root of 4 is 2. But here, they're saying the arithmetic square root of √4. Since √4 is 2, then the arithmetic square root of 2 is √2? Wait, no. Wait, no, √4 is 2, so taking the arithmetic square root of that would just be the same as √4, which is 2. Maybe statement A is just saying that √4 is 2, which is correct. But the wording is a bit confusing. Maybe it's a translation issue? Let me check again. If the question is stating that the arithmetic square root of 4 is 2, then that's correct, but here it's the arithmetic square root of √4. So √4 is 2, and then the arithmetic square root of 2 is √2? Wait, that doesn't make sense. Wait, maybe I'm overcomplicating. Let's parse the sentence again: "The arithmetic square root of √4 is 2". So they are taking the arithmetic square root of the number √4. Since √4 is 2, then the arithmetic square root of 2 is √2, but √2 is approximately 1.414, which is not 2. Wait, that seems contradictory. Alternatively, maybe in Chinese terminology, "arithmetic square root" refers to the principal square root, which is the non-negative one. So for any non-negative number x, the arithmetic square root is the non-negative root. So the arithmetic square root of x is √x. Therefore, if x is √4, which is 2, then the arithmetic square root of √4 is √(√4) = √2 ≈ 1.414. But the statement says it's 2, which would be incorrect. Wait, but hold on, maybe there's confusion in the phrasing. Alternatively, maybe the question is saying that the arithmetic square root of 4 is 2, which would be correct, but the way it's written is "the arithmetic square root of √4 is 2". That seems incorrect. Because √4 is 2, and then the arithmetic square root of 2 is √2. So perhaps statement A is false. Hmm, I need to be careful here. Maybe in some contexts, people refer to arithmetic square root as just the square root, but given the phrasing here, I think statement A is actually incorrect. Because the arithmetic square root of √4 would be the square root of √4, which is the fourth root of 4, which is 2^(2/4) = √2. So that's approximately 1.414. Therefore, statement A is false. Okay, maybe A is wrong.Moving on to option B: The variance of the data 2, 0, 3, 2, 3 is 6/5. Let's calculate that. First, variance. Since this is a population variance or sample variance? The question doesn't specify, but in Chinese curriculum, when they ask for variance without specifying, it's usually population variance, which is divided by N. Let me confirm. The data points are 2, 0, 3, 2, 3. There are 5 data points. First, find the mean. Mean μ = (2 + 0 + 3 + 2 + 3)/5 = (10)/5 = 2. Then variance σ² = [(2-2)² + (0-2)² + (3-2)² + (2-2)² + (3-2)²]/5. Calculating each term: (0)² + (-2)² + (1)² + (0)² + (1)² = 0 + 4 + 1 + 0 + 1 = 6. So variance is 6/5, which is 1.2. So statement B says the variance is 6/5, which is correct. So B is true? But wait, let me check if it's sample variance. If it's sample variance, it would be divided by N-1=4, so 6/4=1.5. But since the answer is 6/5, which is 1.2, that's population variance. Since the problem doesn't specify, but in Chinese exams, unless stated otherwise, variance is usually considered population variance. So B is correct. So B is a true statement.Option C: The sum of interior angles of a regular hexagon is 360°. Wait, the sum of interior angles of a polygon is (n-2)*180°, where n is the number of sides. For a hexagon, n=6. So (6-2)*180 = 4*180 = 720°. So the sum is 720°, not 360°. So statement C is false.Option D: A quadrilateral with diagonals perpendicular to each other is a rhombus. Hmm. Is that true? Wait, a rhombus has perpendicular diagonals, but is the converse true? If a quadrilateral has perpendicular diagonals, is it necessarily a rhombus? No. For example, a kite also has perpendicular diagonals, but a kite is not necessarily a rhombus. A rhombus is a special case of a kite where all sides are equal. So if a quadrilateral has perpendicular diagonals, it could be a kite or a rhombus. So unless we have additional information, like the diagonals bisect each other, then it's a rhombus. But if the diagonals are just perpendicular, it's not necessarily a rhombus. Therefore, statement D is false.So the only true statement is B. Let me just recap:A: Incorrect, because arithmetic square root of √4 is √2, not 2.B: Correct, variance is 6/5.C: Incorrect, sum is 720°.D: Incorrect, could be a kite.Therefore, the answer is B.Final Answerboxed{B}</think>Let's analyze each statement step by step:Statement A: The arithmetic square root of (sqrt{4}) is 2. - The arithmetic square root of 4 is 2, but the question refers to the arithmetic square root of (sqrt{4}). Since (sqrt{4} = 2), the arithmetic square root of 2 is (sqrt{2}), not 2. Therefore, this statement is false.Statement B: The variance of the data 2, 0, 3, 2, 3 is (frac{6}{5}). - Calculate the mean: (mu = frac{2 + 0 + 3 + 2 + 3}{5} = 2). - Calculate the variance: (sigma^2 = frac{(2-2)^2 + (0-2)^2 + (3-2)^2 + (2-2)^2 + (3-2)^2}{5} = frac{0 + 4 + 1 + 0 + 1}{5} = frac{6}{5}). - This statement is true.Statement C: The sum of interior angles of a regular hexagon is (360^circ). - The sum of interior angles of a polygon is ((n-2) times 180^circ). For a hexagon ((n=6)): ((6-2) times 180^circ = 720^circ). - This statement is false.Statement D: A quadrilateral with diagonals perpendicular to each other is a rhombus. - A rhombus has perpendicular diagonals, but a quadrilateral with perpendicular diagonals could be a kite, which is not necessarily a rhombus. - This statement is false.The only true statement is B.(boxed{B}),<think>Alright, let's tackle this problem step by step. So, we have four statements labeled A through D, and we need to figure out which one is true. Let me go through each option one by one.Starting with Option A: "The arithmetic square root of √4 is 2." Hmm, okay. Wait, the term "arithmetic square root" usually refers to the principal (non-negative) square root. So, if we take the square root of 4, the arithmetic square root is indeed 2. But the way it's phrased here is a bit confusing. It says the arithmetic square root of √4 is 2. Wait, √4 is 2, so the arithmetic square root of √4 would be the square root of 2? That doesn't make sense. Wait, maybe I need to parse this correctly.Is it saying that the arithmetic square root of 4 is 2? But they wrote it as √4. So, maybe there's a translation issue here. In Chinese math terminology, "arithmetic square root" is often specified to distinguish from the negative root. So, the square root symbol √ itself typically denotes the arithmetic (principal) square root. So, √4 is 2, which is the arithmetic square root. Therefore, the statement A is essentially saying that the arithmetic square root of 4 is 2, which is correct. But they phrased it as "the arithmetic square root of √4 is 2," which might be redundant or possibly incorrect. Because if you take √4 first, which is 2, and then take the arithmetic square root of that, which is √2, but that's not 2. Wait, that seems conflicting. So maybe there's a misinterpretation here.Hold on, maybe the translation is off. Let me think again. If the statement is saying that "the arithmetic square root of 4 is √4," which is 2, then that's correct. But the way it's written here is "the arithmetic square root of √4 is 2." So, breaking it down: √4 is 2. Then taking the arithmetic square root of that result (which is 2) would be √2 ≈ 1.414, which is not 2. Therefore, statement A is not correct. Hmm, that seems like a trick question. So maybe statement A is false. I need to be careful here.Moving on to Option B: "The variance of the data 2, 0, 3, 2, 3 is 6/5." Let me calculate the variance step by step. First, find the mean of the data set. The numbers are 2, 0, 3, 2, 3. Adding them up: 2 + 0 = 2, 2 + 3 = 5, 5 + 2 = 7, 7 + 3 = 10. So the mean is 10 divided by 5, which is 2.Now, calculate the squared differences from the mean for each data point:- (2 - 2)^2 = 0- (0 - 2)^2 = 4- (3 - 2)^2 = 1- (2 - 2)^2 = 0- (3 - 2)^2 = 1Adding those squared differences: 0 + 4 + 1 + 0 + 1 = 6. Then, the variance is the average of these squared differences. Since this is a population variance (not a sample), we divide by the number of data points, which is 5. So 6 divided by 5 is indeed 6/5. Therefore, statement B seems to be correct. But wait, let me confirm if the question is referring to population variance or sample variance. The term "variance" can sometimes be ambiguous, but in many contexts, especially in exams, unless specified otherwise, it refers to population variance. Since the data set is small and there's no mention of it being a sample, I think it's safe to assume population variance. Therefore, 6/5 is correct. So option B is true? Wait, but let me check once more.Alternatively, if it's sample variance, we would divide by n-1, which is 4, giving 6/4 = 3/2. But since the answer is 6/5, they must be using population variance. Therefore, statement B is correct. Hmm, so that would make B a true statement. But let me hold on that thought and check the other options as well.Option C: "The sum of interior angles of a regular hexagon is 360°." I know that for any polygon, the sum of interior angles can be calculated by the formula (n-2)*180°, where n is the number of sides. A hexagon has 6 sides, so the sum should be (6-2)*180° = 4*180° = 720°. Therefore, the sum is 720°, not 360°. So statement C is false.Option D: "A quadrilateral with diagonals perpendicular to each other is a rhombus." Hmm, okay. A rhombus is a quadrilateral with all sides equal, and its diagonals are perpendicular bisectors of each other. However, is the converse true? If a quadrilateral has perpendicular diagonals, is it necessarily a rhombus?Wait, no. For example, a kite has perpendicular diagonals, but a kite is not necessarily a rhombus unless all its sides are equal. Similarly, a square also has perpendicular diagonals, but it's a special case of a rhombus. However, there are quadrilaterals with perpendicular diagonals that aren't rhombuses. For instance, take a kite where two pairs of adjacent sides are equal but not all four sides. The diagonals are perpendicular, but it's not a rhombus. Therefore, statement D is false.So, going back to the options:- A: Initially confusing, but seems incorrect due to wording. If interpreted as arithmetic square root of 4 is 2, then correct, but as written, arithmetic square root of √4 is 2, which is incorrect because √4 is already 2, then taking square root again would be √2. So A is false.- B: Calculated as 6/5, which matches. So B is true.- C: Sum is 720°, so false.- D: Not necessarily a rhombus, so false.Therefore, the correct answer should be B. But let me double-check my variance calculation once more to be absolutely sure.Data points: 2, 0, 3, 2, 3.Mean: (2+0+3+2+3)/5 = 10/5 = 2.Squared deviations:(2-2)^2 = 0(0-2)^2 = 4(3-2)^2 = 1(2-2)^2 = 0(3-2)^2 = 1Total squared deviations: 0+4+1+0+1=6Population variance: 6/5 = 1.2, which is 6/5. So yes, correct.Therefore, the true statement is B. So I think the answer is B.Final Answerboxed{B}</think>Let's analyze each statement one by one:Option A: The arithmetic square root of (sqrt{4}) is 2. The square root symbol (sqrt{}) denotes the arithmetic (principal) square root. Therefore, (sqrt{4} = 2). However, the statement refers to the arithmetic square root of (sqrt{4}), which would be (sqrt{2}), not 2. Hence, statement A is false.Option B: The variance of the data 2, 0, 3, 2, 3 is (frac{6}{5}). First, calculate the mean: [text{Mean} = frac{2 + 0 + 3 + 2 + 3}{5} = frac{10}{5} = 2] Next, compute the squared differences from the mean: [(2 - 2)^2 = 0, quad (0 - 2)^2 = 4, quad (3 - 2)^2 = 1, quad (2 - 2)^2 = 0, quad (3 - 2)^2 = 1] Sum of squared differences: [0 + 4 + 1 + 0 + 1 = 6] Population variance: [frac{6}{5} = frac{6}{5}] Hence, statement B is true.Option C: The sum of interior angles of a regular hexagon is (360^circ). The formula for the sum of interior angles of an (n)-sided polygon is ((n-2) times 180^circ). For a hexagon ((n = 6)): [(6 - 2) times 180^circ = 4 times 180^circ = 720^circ] Hence, statement C is false.Option D: A quadrilateral with diagonals perpendicular to each other is a rhombus. A quadrilateral with perpendicular diagonals is not necessarily a rhombus (e.g., a kite). Hence, statement D is false.The true statement is (boxed{B}).]

🔑:Let's analyze each statement step-by-step:A: The arithmetic square root of sqrt{4} is 2.- Calculation: sqrt{4} = 2.- However, the arithmetic square root of sqrt{4} is actually sqrt{2}, not 2. Therefore, statement A is boxed{text{False}}.B: The variance of the data 2, 0, 3, 2, 3 is frac{6}{5}.- First, find the mean: frac{2+0+3+2+3}{5} = frac{10}{5} = 2.- Then, calculate the variance: frac{1}{5}[(2-2)^2 + (0-2)^2 + (3-2)^2 + (2-2)^2 + (3-2)^2] = frac{1}{5}[0 + 4 + 1 + 0 + 1] = frac{6}{5}.- Therefore, statement B is boxed{text{True}}.C: The sum of interior angles of a regular hexagon is 360^{circ}.- The formula for the sum of interior angles of a polygon is (n-2) times 180^{circ}, where n is the number of sides.- For a hexagon, n=6, so the sum of interior angles is (6-2) times 180^{circ} = 4 times 180^{circ} = 720^{circ}.- Therefore, statement C is boxed{text{False}}.D: A quadrilateral with diagonals perpendicular to each other is a rhombus.- While it's true that the diagonals of a rhombus are perpendicular, not all quadrilaterals with perpendicular diagonals are rhombuses. The statement lacks the condition that the diagonals also bisect each other at the midpoint, which is a requirement for a quadrilateral to be a rhombus.- Therefore, statement D is boxed{text{False}}.Given the analysis, the correct answer, which matches the question requirements, is boxed{B}.

🧠:[<think>Okay, let's tackle this problem step by step. So, the function is given as f(x) = a - 2/(2^x + 1), and we need to find the value of a such that f(x) is an odd function. Then, we have to find the range of the function and determine its monotonicity over the entire real line. Hmm, let me start with the first part.First, recalling that an odd function satisfies the property f(-x) = -f(x) for all x in its domain. So, if f(x) is odd, then substituting -x into the function should give us the negative of the original function. Let me write that out.Given f(x) = a - 2/(2^x + 1), then f(-x) would be a - 2/(2^{-x} + 1). Let me compute that. 2^{-x} is the same as 1/(2^x), so the denominator becomes 1/(2^x) + 1. Let me combine those terms: 1/(2^x) + 1 = (1 + 2^x)/2^x. Therefore, 2 divided by that would be 2 / [(1 + 2^x)/2^x] = 2 * (2^x)/(1 + 2^x) = (2^{x+1})/(1 + 2^x). Wait, but that's the same as 2*2^x / (1 + 2^x). Let me check that again. Yes, 2 divided by (1/(2^x) + 1) is 2 divided by [(1 + 2^x)/2^x] which is 2 * (2^x)/(1 + 2^x) = 2^{x+1}/(1 + 2^x). So, f(-x) = a - 2^{x+1}/(1 + 2^x).Now, according to the odd function property, f(-x) should equal -f(x). Let's compute -f(x). Since f(x) = a - 2/(2^x + 1), then -f(x) = -a + 2/(2^x + 1). Therefore, setting f(-x) = -f(x):a - [2^{x+1}/(1 + 2^x)] = -a + [2/(2^x + 1)]Let me rearrange terms. Bring all terms to one side:a + a - [2^{x+1}/(1 + 2^x)] - [2/(2^x + 1)] = 0Which simplifies to:2a - [2^{x+1} + 2]/(1 + 2^x) = 0Wait, combining the two fractions:The left side after moving terms becomes 2a = [2^{x+1} + 2]/(1 + 2^x). Let me factor numerator:2^{x+1} is 2*2^x, so numerator is 2*2^x + 2 = 2(2^x + 1). Therefore, [2(2^x + 1)] / (2^x + 1) = 2. Therefore, 2a = 2, which leads to a = 1.Hmm, that seems straightforward. Let me check my steps again to be sure. Starting from f(-x) and -f(x):f(-x) = a - 2/(2^{-x} + 1) = a - [2/( (1/2^x) + 1 )] = a - [2/( (1 + 2^x)/2^x )] = a - [2*(2^x)/(1 + 2^x)] = a - (2^{x+1})/(1 + 2^x)Then, -f(x) = -a + 2/(2^x + 1)Set them equal:a - (2^{x+1})/(1 + 2^x) = -a + 2/(2^x + 1)Bring all terms to the left:a + a - (2^{x+1})/(1 + 2^x) - 2/(2^x + 1) = 0Which is 2a - [2^{x+1} + 2]/(2^x + 1) = 0As I did before, numerator of the fraction is 2^{x+1} + 2 = 2*(2^x + 1). So numerator is 2*(2^x +1), denominator is (2^x +1). Therefore, that fraction is 2. Therefore, 2a - 2 = 0 => a =1. Yep, that checks out. So part 1 answer is a=1.Cool, part 1 done. Now part 2: Find the range of the function y = f(x). Since we now know a=1, the function is f(x) = 1 - 2/(2^x +1). To find the range, we need to see what values y can take as x ranges over all real numbers.Let me express f(x) as 1 - 2/(2^x +1). Let me denote t = 2^x. Since x is a real number, t >0. So, substituting t, we have f(x) = 1 - 2/(t +1). So, let's rewrite this as:y = 1 - 2/(t +1) where t >0.Now, let's analyze the expression 2/(t +1). Since t >0, t +1 >1, so 0 < 1/(t +1) <1. Therefore, 0 < 2/(t +1) <2. Multiplying by -2, but wait, let's see:Wait, the expression is 2/(t +1). Since t +1 >1, 2/(t +1) <2. Also, as t approaches infinity, t +1 approaches infinity, so 2/(t +1) approaches 0. When t approaches 0 from the right, t +1 approaches 1, so 2/(t +1) approaches 2. Therefore, 2/(t +1) ranges from 0 to 2, not including 0 and 2. Therefore, when we subtract that from 1, we get:1 - (something between 0 and 2) = (1 - 2) to (1 - 0) = -1 to 1. But since 2/(t +1) is strictly between 0 and 2, the range of y is (-1,1). Hmm, but let's verify that.Wait, t can be any positive real number. Let's write y = 1 - 2/(t +1). Let's solve for t in terms of y:y = 1 - 2/(t +1)So, subtract 1: y -1 = -2/(t +1)Multiply both sides by -1: 1 - y = 2/(t +1)Therefore, t +1 = 2/(1 - y)Hence, t = 2/(1 - y) -1 = [2 - (1 - y)]/(1 - y) = (1 + y)/(1 - y)But since t >0, (1 + y)/(1 - y) >0So, the fraction (1 + y)/(1 - y) is positive. Therefore, the numerator and denominator must have the same sign.Case 1: Both numerator and denominator are positive.1 + y >0 => y > -11 - y >0 => y <1So, -1 < y <1Case 2: Both numerator and denominator are negative.1 + y <0 => y < -11 - y <0 => y >1But these two can't be true at the same time because y can't be both less than -1 and greater than 1. So, only possible case is -1 < y <1. Therefore, the range of y is (-1,1).But wait, let me check if y can actually reach -1 or 1. As t approaches 0, which is the lower bound of t (since t>0), then y approaches 1 - 2/(0 +1) = 1 -2 = -1. Wait, that's contradictory. Wait, if t approaches 0 from the right, then t approaches 0, so 2/(t +1) approaches 2/(0 +1)=2. So, y approaches 1 -2= -1. Similarly, as t approaches infinity, 2/(t +1) approaches 0, so y approaches 1 -0=1. Therefore, y approaches -1 and 1, but never actually reaches them. Therefore, the range is (-1,1). So part 2 answer is (-1,1).Hmm, that seems right. Let me verify with an example. For x=0, t=1, y=1 -2/(1+1)=1 -1=0. For x approaching infinity, t approaches infinity, so y approaches 1. For x approaching negative infinity, t approaches 0, so y approaches -1. So y ranges between -1 and1, open interval. So yes, the range is (-1,1).Now part 3: Determine the monotonicity of f(x) on (-infty, +infty) and prove using the definition. So, monotonicity refers to whether the function is increasing or decreasing. Let's first try to get an intuition. Since f(x) =1 - 2/(2^x +1). Let's see, as x increases, 2^x increases, so 2/(2^x +1) decreases, so subtracting that from 1 means the whole expression increases. Therefore, f(x) is increasing. To confirm, let's check the derivative, but the problem says to use the definition, so perhaps we need to use the definition of monotonicity.But maybe since the problem allows proving using the definition, perhaps we can consider two arbitrary points x1 < x2 and show that f(x1) < f(x2). Alternatively, compute the difference f(x2) - f(x1) and show it's positive.Alternatively, maybe using the derivative is allowed if the problem says "using the definition" but perhaps they mean the definition of increasing function. Let me check.Definition: A function f is increasing on an interval if for any x1 < x2 in the interval, f(x1) < f(x2).Alternatively, for differentiable functions, if the derivative is positive everywhere, the function is increasing. But the problem says "using the definition", so perhaps they want the first approach.But given that calculus might not be necessary here, even though the problem mentions "using the definition", maybe we can use algebraic manipulation. Let me see.Alternatively, maybe transform the function into a more familiar form. Let's write f(x) =1 - 2/(2^x +1). Let's see, 2/(2^x +1) can be rewritten as 2/(2^x +1). Alternatively, maybe express this as 1 - 2/(2^x +1). Let's compute f(x2) - f(x1):[1 - 2/(2^{x2} +1)] - [1 - 2/(2^{x1} +1)] = -2/(2^{x2} +1) +2/(2^{x1} +1) = 2[1/(2^{x1} +1) -1/(2^{x2} +1)]So, f(x2) - f(x1) = 2[ (2^{x2} +1 -2^{x1} -1)/ ( (2^{x1} +1)(2^{x2} +1) ) ] = 2[ (2^{x2} -2^{x1}) / ( (2^{x1} +1)(2^{x2} +1) ) ]So, factor out 2^{x1} from the numerator:= 2[2^{x1}(2^{x2 -x1} -1) / ( (2^{x1} +1)(2^{x2} +1) ) ]Now, if x2 >x1, then exponent (x2 -x1) >0, so 2^{x2 -x1} >1, so numerator is positive. Denominator is positive as 2^{x1} +1 and 2^{x2} +1 are both positive. Therefore, entire expression is positive. Therefore, f(x2) - f(x1) >0 when x2 >x1, so f is increasing.Therefore, the function is strictly increasing on (-infty, +infty).Alternatively, maybe there's a simpler way. Let me see. Let's suppose x1 <x2. Since 2^x is an increasing function, 2^{x1} <2^{x2}. Then, 2^{x1} +1 <2^{x2} +1. Therefore, 1/(2^{x1} +1) >1/(2^{x2} +1). Multiplying by 2, 2/(2^{x1} +1) >2/(2^{x2} +1). Therefore, -2/(2^{x1} +1) < -2/(2^{x2} +1). Adding 1 to both sides, 1 -2/(2^{x1} +1) <1 -2/(2^{x2} +1). Therefore, f(x1) <f(x2). Therefore, f is strictly increasing.Yes, that works. So, using the fact that 2^x is increasing, and then reciprocal reverses inequality, so 2/(2^x +1) is decreasing, so negative of that is increasing, adding 1 doesn't affect the monotonicity. Therefore, f(x) is strictly increasing.So the conclusion is that f(x) is strictly increasing on (-infty, +infty), and the proof can be done by either algebraic manipulation as above or by using properties of increasing/decreasing functions. But according to the problem statement, we need to prove it using the definition, so probably the approach where we take x1 <x2 and show f(x1) <f(x2) is appropriate.Therefore, all parts solved:1. a=12. Range is (-1,1)3. f(x) is strictly increasing on (-infty, +infty), proven by the definition.Final Answer1. The value of ( a ) is boxed{1}.2. The range of the function ( y = f(x) ) is (boxed{(-1, 1)}).3. The function ( f(x) ) is strictly increasing on ((-infty, +infty)).</think>Given the function ( f(x) = a - frac{2}{2^x + 1} ), where ( a in mathbb{R} ), and ( f(x) ) is an odd function:1. Finding the value of ( a ): - An odd function satisfies ( f(-x) = -f(x) ). - Compute ( f(-x) ): [ f(-x) = a - frac{2}{2^{-x} + 1} = a - frac{2 cdot 2^x}{1 + 2^x} ] - Set ( f(-x) = -f(x) ): [ a - frac{2 cdot 2^x}{1 + 2^x} = -a + frac{2}{1 + 2^x} ] - Simplifying, we find: [ 2a = 2 implies a = 1 ] - Thus, the value of ( a ) is (boxed{1}).2. Finding the range of the function ( y = f(x) ): - With ( a = 1 ), the function becomes ( f(x) = 1 - frac{2}{2^x + 1} ). - Let ( t = 2^x ) where ( t > 0 ). Then: [ y = 1 - frac{2}{t + 1} ] - Analyze the expression ( frac{2}{t + 1} ): - As ( t ) approaches 0, ( frac{2}{t + 1} ) approaches 2, so ( y ) approaches -1. - As ( t ) approaches infinity, ( frac{2}{t + 1} ) approaches 0, so ( y ) approaches 1. - Therefore, the range of ( y ) is ((-1, 1)). - The range of the function ( y = f(x) ) is (boxed{(-1, 1)}).3. Determining the monotonicity of ( f(x) ): - To show ( f(x) ) is strictly increasing, consider ( x_1 < x_2 ): - Since ( 2^x ) is increasing, ( 2^{x_1} < 2^{x_2} ). - Thus, ( frac{2}{2^{x_1} + 1} > frac{2}{2^{x_2} + 1} ). - Therefore, ( 1 - frac{2}{2^{x_1} + 1} < 1 - frac{2}{2^{x_2} + 1} ), implying ( f(x_1) < f(x_2) ). - Hence, ( f(x) ) is strictly increasing on ((-infty, +infty)). - The function ( f(x) ) is strictly increasing on ((-infty, +infty)).Final Answers:1. The value of ( a ) is (boxed{1}).2. The range of the function ( y = f(x) ) is (boxed{(-1, 1)}).3. The function ( f(x) ) is strictly increasing on ((-infty, +infty)).,<think>Okay, let's tackle this problem step by step. The function given is ( f(x) = a - frac{2}{2^x + 1} ), and we know it's an odd function. There are three parts to the problem: finding the value of ( a ), finding the range of ( f(x) ), and determining the monotonicity of ( f(x) ) over all real numbers. Let's start with the first part.1. Finding the value of ( a ):Since ( f(x) ) is an odd function, it must satisfy the condition ( f(-x) = -f(x) ) for all ( x ). Let's compute ( f(-x) ) and set it equal to ( -f(x) ), then solve for ( a ).First, compute ( f(-x) ):[f(-x) = a - frac{2}{2^{-x} + 1} = a - frac{2}{frac{1}{2^x} + 1}]Let me simplify the denominator here. The denominator is ( frac{1}{2^x} + 1 ), which can be written as ( frac{1 + 2^x}{2^x} ). Therefore, the entire expression becomes:[a - frac{2}{frac{1 + 2^x}{2^x}} = a - 2 cdot frac{2^x}{1 + 2^x} = a - frac{2^{x+1}}{1 + 2^x}]Wait, that seems a bit complicated. Let me check my steps again. Starting from ( f(-x) ):[f(-x) = a - frac{2}{2^{-x} + 1} = a - frac{2}{frac{1}{2^x} + 1}]To combine the terms in the denominator:[frac{1}{2^x} + 1 = frac{1 + 2^x}{2^x}]So, taking the reciprocal and multiplying by 2:[frac{2}{frac{1 + 2^x}{2^x}} = 2 cdot frac{2^x}{1 + 2^x} = frac{2 cdot 2^x}{1 + 2^x} = frac{2^{x+1}}{1 + 2^x}]Therefore, ( f(-x) = a - frac{2^{x+1}}{1 + 2^x} ).On the other hand, ( -f(x) ) is:[-f(x) = -left( a - frac{2}{2^x + 1} right) = -a + frac{2}{2^x + 1}]Since ( f(-x) = -f(x) ), we set them equal:[a - frac{2^{x+1}}{1 + 2^x} = -a + frac{2}{2^x + 1}]Let's rearrange terms to solve for ( a ). Bring all terms to one side:[a + a = frac{2^{x+1}}{1 + 2^x} + frac{2}{2^x + 1}]Simplify the left side:[2a = frac{2^{x+1} + 2}{1 + 2^x}]Factor numerator on the right:[2a = frac{2(2^x + 1)}{1 + 2^x} = frac{2(1 + 2^x)}{1 + 2^x} = 2]Therefore:[2a = 2 implies a = 1]So, the value of ( a ) is 1. Let me verify this result to make sure. If ( a = 1 ), then the function is:[f(x) = 1 - frac{2}{2^x + 1}]Let's check if it's odd. Compute ( f(-x) ):[f(-x) = 1 - frac{2}{2^{-x} + 1} = 1 - frac{2}{frac{1}{2^x} + 1} = 1 - frac{2 cdot 2^x}{1 + 2^x} = 1 - frac{2^{x+1}}{1 + 2^x}]Now, compute ( -f(x) ):[-f(x) = -1 + frac{2}{2^x + 1}]Let's see if ( f(-x) = -f(x) ):Left-hand side (LHS):[1 - frac{2^{x+1}}{1 + 2^x} = 1 - frac{2 cdot 2^x}{1 + 2^x}]Right-hand side (RHS):[-1 + frac{2}{1 + 2^x}]Hmm, let's compute LHS - RHS to check:[1 - frac{2 cdot 2^x}{1 + 2^x} + 1 - frac{2}{1 + 2^x} = 2 - frac{2 cdot 2^x + 2}{1 + 2^x}]Simplify numerator in the fraction:[2 cdot 2^x + 2 = 2(2^x + 1)]So, the expression becomes:[2 - frac{2(2^x + 1)}{1 + 2^x} = 2 - 2 = 0]Therefore, LHS - RHS = 0, which means LHS = RHS. Hence, when ( a = 1 ), ( f(-x) = -f(x) ), so the function is indeed odd. Therefore, part 1 is solved: ( a = 1 ).2. Finding the range of the function ( y = f(x) ):Now that we have ( a = 1 ), the function becomes:[f(x) = 1 - frac{2}{2^x + 1}]To find the range, we need to determine all possible values ( y ) can take. Let's analyze the function step by step.First, consider the term ( frac{2}{2^x + 1} ). Let's denote ( t = 2^x ). Since ( 2^x > 0 ) for all real ( x ), ( t ) can take any positive real number: ( t in (0, +infty) ).So, substituting ( t ), the function becomes:[f(t) = 1 - frac{2}{t + 1}]Now, let's analyze ( frac{2}{t + 1} ). As ( t ) approaches 0 from the right, ( t + 1 ) approaches 1, so ( frac{2}{t + 1} ) approaches 2. As ( t ) approaches infinity, ( t + 1 ) approaches infinity, so ( frac{2}{t + 1} ) approaches 0. Therefore, ( frac{2}{t + 1} ) takes values in ( (0, 2) ).Therefore, ( f(t) = 1 - frac{2}{t + 1} ). Let's see the range of ( f(t) ).Since ( frac{2}{t + 1} in (0, 2) ), subtracting this from 1 gives:- When ( frac{2}{t + 1} ) approaches 2, ( f(t) ) approaches ( 1 - 2 = -1 ).- When ( frac{2}{t + 1} ) approaches 0, ( f(t) ) approaches ( 1 - 0 = 1 ).But we need to check if all values between -1 and 1 are covered. Since ( t ) is continuous over ( (0, +infty) ), the function ( frac{2}{t + 1} ) is continuous, so ( f(t) ) is also continuous. Therefore, as ( t ) moves from 0 to infinity, ( f(t) ) moves from -1 to 1, covering all values in between. However, we need to confirm if the endpoints are included or excluded.Since ( t ) is never actually 0 or infinity, ( frac{2}{t + 1} ) never actually reaches 2 or 0. Thus, ( f(t) ) never actually reaches -1 or 1. Therefore, the range of ( f(t) ) is ( (-1, 1) ).But let's verify this with some test values. Let me plug in some ( x ) values:- When ( x ) approaches ( -infty ), ( 2^x ) approaches 0, so ( frac{2}{2^x + 1} ) approaches ( 2/(0 + 1) = 2 ), so ( f(x) ) approaches ( 1 - 2 = -1 ).- When ( x = 0 ), ( 2^0 = 1 ), so ( frac{2}{1 + 1} = 1 ), so ( f(0) = 1 - 1 = 0 ).- When ( x ) approaches ( +infty ), ( 2^x ) approaches infinity, so ( frac{2}{2^x + 1} ) approaches 0, so ( f(x) ) approaches ( 1 - 0 = 1 ).But even though ( x = 0 ) gives ( f(x) = 0 ), which is within (-1, 1). However, the endpoints -1 and 1 are approached but not achieved. Therefore, the range of ( f(x) ) is ( (-1, 1) ).Wait, but the problem says "the range of the function ( y = f(x) )". So, the answer should be the open interval from -1 to 1.Alternatively, maybe there's a different approach. Let's express ( y = 1 - frac{2}{2^x + 1} ) and solve for ( 2^x ) in terms of ( y ).Starting with:[y = 1 - frac{2}{2^x + 1}]Subtract 1 from both sides:[y - 1 = - frac{2}{2^x + 1}]Multiply both sides by -1:[1 - y = frac{2}{2^x + 1}]Then, solve for ( 2^x ):[2^x + 1 = frac{2}{1 - y}][2^x = frac{2}{1 - y} - 1 = frac{2 - (1 - y)}{1 - y} = frac{1 + y}{1 - y}]Therefore:[2^x = frac{1 + y}{1 - y}]Since ( 2^x > 0 ) for all real ( x ), the right-hand side must also be positive. So:[frac{1 + y}{1 - y} > 0]This inequality holds when both numerator and denominator are positive or both are negative.Case 1: Both numerator and denominator positive:- ( 1 + y > 0 implies y > -1 )- ( 1 - y > 0 implies y < 1 )So, combining these: ( -1 < y < 1 )Case 2: Both numerator and denominator negative:- ( 1 + y < 0 implies y < -1 )- ( 1 - y < 0 implies y > 1 )But these inequalities can't be satisfied simultaneously because ( y < -1 ) and ( y > 1 ) can't happen at the same time. So, no solution in this case.Therefore, the only valid solutions are when ( -1 < y < 1 ). Hence, the range of ( f(x) ) is ( (-1, 1) ). So part 2 is solved.3. Determining the monotonicity of ( f(x) ) on ( (-infty, +infty) ) and proving it using the definition.We need to determine if the function is increasing or decreasing over the entire real line. Let's first analyze the function ( f(x) = 1 - frac{2}{2^x + 1} ).Intuitively, since ( 2^x ) is an increasing function, the denominator ( 2^x + 1 ) is increasing, so ( frac{1}{2^x + 1} ) is decreasing, which makes ( frac{2}{2^x + 1} ) decreasing as well. Therefore, ( f(x) = 1 - frac{2}{2^x + 1} ) would be increasing, because subtracting a decreasing function results in an increasing function.But to prove this formally, we need to use the definition of monotonicity. The problem specifies to use the definition, which typically involves showing that for any ( x_1 < x_2 ), ( f(x_1) < f(x_2) ) (if increasing) or ( f(x_1) > f(x_2) ) (if decreasing).Alternatively, we can compute the derivative, but since the problem says "using the definition," perhaps we need to proceed without calculus. Let me check.Wait, the problem says "determine the monotonicity... and prove your conclusion using the definition." The definition here is likely the standard definition of an increasing or decreasing function. Let's recall:A function ( f ) is increasing on an interval if for any ( x_1 < x_2 ) in that interval, ( f(x_1) leq f(x_2) ). If the inequality is strict, it's strictly increasing. Similarly for decreasing.Since we need to determine strict monotonicity here, given the function is likely strictly increasing.But let's try to do this step by step. Let's take two arbitrary points ( x_1 ) and ( x_2 ) with ( x_1 < x_2 ), and compare ( f(x_1) ) and ( f(x_2) ).Given ( x_1 < x_2 ), since ( 2^x ) is increasing, ( 2^{x_1} < 2^{x_2} ).So, ( 2^{x_1} + 1 < 2^{x_2} + 1 ), therefore ( frac{1}{2^{x_1} + 1} > frac{1}{2^{x_2} + 1} ). Multiplying both sides by 2 (positive number preserves inequality), ( frac{2}{2^{x_1} + 1} > frac{2}{2^{x_2} + 1} ).Then, ( -frac{2}{2^{x_1} + 1} < -frac{2}{2^{x_2} + 1} ).Adding 1 to both sides:[1 - frac{2}{2^{x_1} + 1} < 1 - frac{2}{2^{x_2} + 1}]Which is ( f(x_1) < f(x_2) ).Therefore, if ( x_1 < x_2 ), then ( f(x_1) < f(x_2) ), so ( f(x) ) is strictly increasing on ( (-infty, +infty) ).Alternatively, using calculus: compute the derivative.But the problem specifies using the definition. However, perhaps the above reasoning uses the property of the function's components, but is that considered "using the definition"? The step-by-step comparison of ( x_1 < x_2 ) leading to ( f(x_1) < f(x_2) ) is the standard way to prove monotonicity via the definition.Alternatively, maybe we need to show more directly. Let me try to frame it more formally.Let ( x_1 < x_2 ). We need to show that ( f(x_2) - f(x_1) > 0 ).Compute ( f(x_2) - f(x_1) ):[left[1 - frac{2}{2^{x_2} + 1}right] - left[1 - frac{2}{2^{x_1} + 1}right] = frac{2}{2^{x_1} + 1} - frac{2}{2^{x_2} + 1}]Factor out the 2:[2 left( frac{1}{2^{x_1} + 1} - frac{1}{2^{x_2} + 1} right)]Combine the fractions:[2 left( frac{2^{x_2} + 1 - (2^{x_1} + 1)}{(2^{x_1} + 1)(2^{x_2} + 1)} right) = 2 left( frac{2^{x_2} - 2^{x_1}}{(2^{x_1} + 1)(2^{x_2} + 1)} right)]Since ( x_1 < x_2 ), ( 2^{x_2} - 2^{x_1} > 0 ). Also, the denominator ( (2^{x_1} + 1)(2^{x_2} + 1) ) is positive because both factors are positive. Therefore, the entire expression is positive. Hence, ( f(x_2) - f(x_1) > 0 ), so ( f(x) ) is strictly increasing on ( mathbb{R} ).Therefore, part 3: ( f(x) ) is strictly increasing on ( (-infty, +infty) ), proven by the definition.Summary of solutions:1. ( a = 1 )2. The range is ( (-1, 1) )3. ( f(x) ) is strictly increasing on ( (-infty, +infty) ), proven by comparing ( f(x_2) - f(x_1) > 0 ) for ( x_1 < x_2 ).Final Answer1. (boxed{1}) 2. (boxed{(-1, 1)}) 3. The function ( f(x) ) is strictly increasing on ((-infty, +infty)).</think>Given the function ( f(x) = a - frac{2}{2^x + 1} ), where ( a in mathbb{R} ), and ( f(x) ) is an odd function:1. Finding the value of ( a ): Since ( f(x) ) is an odd function, it must satisfy ( f(-x) = -f(x) ). By computing ( f(-x) ) and setting it equal to ( -f(x) ), we solve for ( a ): [ f(-x) = a - frac{2}{2^{-x} + 1} = a - frac{2 cdot 2^x}{1 + 2^x} ] [ -f(x) = -a + frac{2}{2^x + 1} ] Equating and solving, we find: [ 2a = 2 implies a = 1 ] Thus, the value of ( a ) is (boxed{1}).2. Finding the range of the function ( y = f(x) ): With ( a = 1 ), the function becomes ( f(x) = 1 - frac{2}{2^x + 1} ). By analyzing the term ( frac{2}{2^x + 1} ), we find it takes values in ( (0, 2) ). Therefore, ( f(x) ) takes values in ( (-1, 1) ). Solving for ( y ) in terms of ( 2^x ), we confirm the range is ( (-1, 1) ). Hence, the range is (boxed{(-1, 1)}).3. Determining the monotonicity of ( f(x) ): For ( x_1 < x_2 ), we show ( f(x_1) < f(x_2) ): [ f(x_2) - f(x_1) = 2 left( frac{2^{x_2} - 2^{x_1}}{(2^{x_1} + 1)(2^{x_2} + 1)} right) ] Since ( 2^{x_2} - 2^{x_1} > 0 ) and the denominator is positive, ( f(x) ) is strictly increasing. Therefore, the function ( f(x) ) is strictly increasing on ((-infty, +infty)). Final Answer1. (boxed{1}) 2. (boxed{(-1, 1)}) 3. The function ( f(x) ) is strictly increasing on ((-infty, +infty)).]

🔑:Solution:1. Given the function, we have f(x) = frac{a cdot 2^x + a - 2}{2^x + 1}.Since f(x) is an odd function, we have f(-x) = -f(x), which means frac{a cdot 2^{-x} + a - 2}{2^{-x} + 1} = -frac{a cdot 2^x + a - 2}{2^x + 1}. Therefore, we get a - 2 = -a, which implies a = 1. Thus, f(x) = 1 - frac{2}{2^x + 1}; (3 points)2. Since f(x) = 1 - frac{2}{2^x + 1}, and 2^x + 1 > 1, we have frac{2}{2^x + 1} in (0, 2). Therefore, the range of f(x) is (-1, 1); (6 points)3. Let x_1, x_2 be any two values in the interval (-infty, +infty), with x_1 < x_2. Then, 0 < 2^{x_1} < 2^{x_2}, and 2^{x_1} - 2^{x_2} < 0. Since f(x_1) - f(x_2) = frac{2(2^{x_1} - 2^{x_2})}{(2^{x_1} + 1)(2^{x_2} + 1)} < 0, we have f(x_1) < f(x_2). Therefore, f(x) is an increasing function on (-infty, +infty). (12 points)Thus, the answers are:1. a = boxed{1}2. The range of f(x) is boxed{(-1, 1)}3. f(x) is an increasing function on (-infty, +infty).

🧠:[<think>Alright, so I need to find the length of the arc MP in this semicircle. Let me first visualize the problem. There's a semicircle with center O. Inside it, there's a shaded closed figure formed by two parallel chords MN and PQ. MN is 6 meters long, PQ is 8 meters long, and the distance between these two chords is 2 meters. The question is asking for the length of the arc MP, accurate to three decimal places.Hmm, okay. Since MN and PQ are parallel chords in the semicircle, their distances from the center O must be different. The arc length MP depends on the angle it subtends at the center. So if I can find the angle ∠MOP, then the arc length would be (θ/360) * 2πr, but since it's a semicircle, the circumference is πr. Wait, no, the semicircle circumference is πr, so the full circle would be 2πr. So the arc length MP would be (θ/360) * 2πr, where θ is the central angle in degrees. But maybe it's easier to work in radians, where the arc length is simply rθ, with θ in radians.First, I need to find the radius of the semicircle. Let's denote the radius as r. Since MN and PQ are chords of the semicircle, their lengths are related to their distances from the center. Let me recall the formula for the length of a chord: if a chord of length l is at distance d from the center of a circle with radius r, then l = 2√(r² - d²). So, for chord MN (6m) and chord PQ (8m), their distances from the center can be calculated if we know the radius. Let me denote the distance from the center O to MN as h, and the distance from O to PQ as k. Then, since MN and PQ are parallel and the distance between them is 2m, the distance between h and k is 2m. However, depending on their positions relative to the center, the total distance between the chords could be h + k or |h - k|. Since MN and PQ are both in the semicircle, they must both be on the same side of the diameter. Wait, in a semicircle, the diameter is the base. So MN and PQ are both above the diameter? Wait, but in a semicircle, the chords are all above the diameter. Wait, no. If the semicircle is, say, the upper half of a circle with diameter AB, then all chords are within the upper half. But the problem says the shaded part is a closed figure within the semicircle, so MN and PQ are two chords parallel to each other, and the figure is between them. The distance between MN and PQ is 2m. Since they're chords in the semicircle, their distances from the center must be such that the distance between them is 2m.But since the chords are parallel, their distances from the center can be calculated. Let me suppose that chord MN is closer to the center than chord PQ. Then, the distance from O to MN is h, and the distance from O to PQ is h + 2. But wait, if the distance between the two chords is 2m, then depending on which side of the center they are, the distances could be h and h + 2, or h and h - 2. But in a semicircle, all chords must be on one side of the diameter. Wait, perhaps MN and PQ are both above the diameter? Wait, the semicircle itself is the top half. So the diameter is the base, and the center O is the midpoint of the diameter. Then, the chords MN and PQ are both above the diameter, at different heights. The distance between the chords is 2m. So the vertical distance between MN and PQ is 2m. So, if I take the distance from O to MN as h, then the distance from O to PQ is h + 2 or h - 2. But since PQ is longer than MN (8m vs 6m), PQ must be closer to the center. Because in a circle, longer chords are closer to the center. So PQ is longer, so it's closer to the center. Therefore, if MN is 6m and PQ is 8m, then PQ is closer to O. Therefore, the distance from O to PQ is less than the distance from O to MN. Wait, but the distance between the chords is given as 2m. So if O is the center, and PQ is closer to O, then MN is further away. So the vertical distance between PQ and MN is 2m, with MN being 2m further from O than PQ. Let's denote the distance from O to PQ as h, then the distance from O to MN is h + 2. Then, using the chord length formula:For PQ (8m): 8 = 2√(r² - h²)For MN (6m): 6 = 2√(r² - (h + 2)²)Let me write those equations out:8 = 2√(r² - h²) ⇒ √(r² - h²) = 4 ⇒ r² - h² = 16 ...(1)6 = 2√(r² - (h + 2)²) ⇒ √(r² - (h + 2)²) = 3 ⇒ r² - (h + 2)² = 9 ...(2)So now, subtract equation (2) from equation (1):[r² - h²] - [r² - (h + 2)²] = 16 - 9 ⇒ -h² + (h + 2)² = 7Expanding (h + 2)²: h² + 4h + 4. Therefore:(-h²) + (h² + 4h + 4) = 7 ⇒ 4h + 4 = 7 ⇒ 4h = 3 ⇒ h = 3/4 = 0.75 mThen, substitute h = 0.75 into equation (1):r² - (0.75)² = 16 ⇒ r² = 16 + 0.5625 = 16.5625 ⇒ r = √16.5625 = 4.0625 mWait, √16.5625. Let me compute that. 4.0625 squared: 4^2 = 16, 0.0625*16 = 1, so 4.0625^2 = 16 + 2*4*0.0625 + 0.0625^2 = 16 + 0.5 + 0.00390625 = 16.50390625. Wait, that's not matching. Wait, actually, 4.0625 is equal to 4 + 0.0625, which is 4 + 1/16. So (4 + 1/16)^2 = 16 + 2*(4)*(1/16) + (1/16)^2 = 16 + 0.5 + 1/256 ≈ 16.50390625. But according to equation (1), r² = 16.5625. Wait, 16.5625 is 16 + 9/16, since 9/16 is 0.5625. So √(16.5625) is 4.0625? Wait, 4.0625 squared is 16.50390625, which is not 16.5625. So maybe my calculation is wrong.Wait, let me check again. If h = 0.75, then equation (1):r² = 16 + h² = 16 + (0.75)^2 = 16 + 0.5625 = 16.5625. Therefore, r = sqrt(16.5625). Let's compute sqrt(16.5625). Since 4^2 = 16, and 4.1^2 = 16.81, which is higher than 16.5625. So 16.5625 - 16 = 0.5625. So sqrt(16.5625) = 4 + x, where (4 + x)^2 = 16 + 8x + x² = 16.5625. So 8x + x² = 0.5625. Let's approximate x. If x=0.07, 8*0.07=0.56, x²=0.0049, total 0.5649 ≈ 0.5625. Close enough. So x≈0.07. So sqrt(16.5625)≈4.07. But actually, 4.0625^2 = (4 + 0.0625)^2 = 16 + 2*4*0.0625 + 0.0625^2 = 16 + 0.5 + 0.00390625 = 16.50390625. Hmm, not 16.5625. Wait, maybe I made a mistake in solving the equations.Wait, let's step back. Equations (1) and (2):1) r² - h² = 162) r² - (h + 2)^2 = 9Subtract equation (2) from equation (1):[r² - h²] - [r² - (h + 2)^2] = 16 - 9Simplify:-r² + h² + r² - (h² + 4h + 4) = 7Wait, no. Wait, expanding the subtraction:(r² - h²) - (r² - (h + 2)^2) = 16 - 9Which is:r² - h² - r² + (h + 2)^2 = 7Simplify:- h² + (h² + 4h + 4) = 7Which simplifies to:4h + 4 = 7So 4h = 3 ⇒ h = 0.75. So that part is correct.So substituting h = 0.75 into equation (1):r² = 16 + (0.75)^2 = 16 + 0.5625 = 16.5625Therefore, r = sqrt(16.5625). Let's compute this exactly. 16.5625 * 16 = 265. So sqrt(265/16) = sqrt(265)/4. sqrt(265) is approximately 16.2788. So 16.2788/4 ≈ 4.0697. Hmm, but 4.0625 squared is 16.5039 as before, which is less than 16.5625. So perhaps 4.0697 squared is approximately 16.5625. Let's check:4.0697^2 = (4 + 0.0697)^2 = 16 + 2*4*0.0697 + 0.0697^2 ≈ 16 + 0.5576 + 0.004858 ≈ 16.5624, which is close enough. So r ≈ 4.0697 meters.So the radius is approximately 4.0697 meters.Now, we need to find the length of the arc MP. To find this, we need to determine the central angle ∠MOP, then multiply it by the radius. So first, we need to find the angle corresponding to arc MP.But where exactly are points M, N, P, Q located? Let me visualize the semicircle. Let's assume the semicircle is drawn with diameter AB horizontally, center O at the midpoint. The two parallel chords MN and PQ are also horizontal (since they are parallel to each other and presumably parallel to the diameter AB). Chord PQ is 8 meters long and closer to the center O, at a distance h = 0.75 meters above AB (assuming AB is the diameter at the base). Chord MN is 6 meters long and is 2 meters above PQ, so its distance from O is h + 2 = 0.75 + 2 = 2.75 meters above AB? Wait, no. Wait, if PQ is at distance h from O, and MN is 2 meters away from PQ, then since PQ is closer to the center, MN is 2 meters further away from the center. Wait, the distance between the chords is 2 meters. So the vertical distance between PQ and MN is 2 meters.But since PQ is at distance h from O, and MN is at distance h + 2 from O? Wait, no, because the distance from O to PQ is h, and the vertical distance between PQ and MN is 2 meters. Since MN is above PQ (if the semicircle is above diameter AB), then the distance from O to MN would be h + 2. Wait, but in the semicircle, the maximum distance from O is the radius, which is approximately 4.0697 meters. If h is 0.75, then h + 2 = 2.75, which is less than the radius. That's okay.But to find the arc MP, we need to find points M and P on the semicircle. Let me assume that points M and P are endpoints of the arcs above the chords MN and PQ. Wait, actually, since MN and PQ are chords, each chord has two endpoints. So chord MN has endpoints M and N, chord PQ has endpoints P and Q. Since the figure is a closed figure within the semicircle, the shaded area is probably the region between the two chords MN and PQ and the two arcs connecting them. Therefore, the arc MP would be one of the arcs connecting M to P along the semicircle.But how are points M, N, P, Q arranged? Since the chords are parallel, and the figure is closed, likely the endpoints are connected by arcs. So starting from M, going along the semicircle to P, then down along PQ to Q, then back along the semicircle to N, and then back along MN to M? Or maybe not. Wait, the closed figure is within the semicircle, so maybe it's bounded by the two chords and the two arcs MP and NQ? But the problem mentions finding the length of arc MP, so we need to find the arc from M to P.Assuming that M and P are adjacent endpoints of the two chords. Let me try to sketch this mentally. The semicircle has center O. Chord PQ is closer to O, length 8m, at distance h = 0.75m from O. Chord MN is 6m, at distance h + 2 = 2.75m from O. Let's assume that chord PQ is above chord MN (but since PQ is closer to the center, which is at the top of the semicircle? Wait, no. Wait, in a semicircle drawn with diameter AB at the bottom, center O is at the midpoint of AB. Then, the semicircle is the upper half. So moving up from AB, the center O is at the base. Wait, no, the center O is at the midpoint of AB, which is the diameter. So in the semicircle, the highest point is directly above O, at a distance equal to the radius. So the top of the semicircle is at point C, which is r meters above O.Wait, maybe I'm overcomplicating. Let's model this with coordinates. Let's set up a coordinate system where the diameter AB is along the x-axis from (-r, 0) to (r, 0), center O at (0,0), and the semicircle is the upper half with equation x² + y² = r², y ≥ 0.Then, chord PQ is parallel to the x-axis (since it's parallel to MN and the diameter AB is the x-axis). Since PQ is a chord of length 8m, located at a distance h = 0.75m from the center O. Wait, but distance from the center is along the y-axis. So chord PQ is horizontal, at y = h = 0.75m. The length of PQ is 8m. So in the coordinate system, chord PQ goes from point (-4, 0.75) to (4, 0.75), since the length is 8m. Wait, no. Wait, in the circle equation x² + y² = r², a horizontal chord at y = 0.75 would have length 2√(r² - 0.75²). But we know that PQ is 8m, so:8 = 2√(r² - h²) ⇒ 4 = √(r² - 0.75²) ⇒ 16 = r² - 0.5625 ⇒ r² = 16.5625 ⇒ r = √16.5625 ≈ 4.0697m.So chord PQ is at y = 0.75, extending from x = -4 to x = 4? Wait, but 2√(r² - h²) = 8. So √(r² - h²) = 4 ⇒ half the chord length is 4. So chord PQ goes from (-4, 0.75) to (4, 0.75). Similarly, chord MN is at y = 0.75 + 2 = 2.75, but wait, the distance between PQ and MN is 2m. Since they are parallel chords, the vertical distance between them is 2m. So if PQ is at y = 0.75, then MN is at y = 0.75 + 2 = 2.75. However, we need to check if that's within the semicircle. The radius is approximately 4.0697, so the highest point of the semicircle is at y = r ≈ 4.0697, so 2.75 is less than that, which is fine.Chord MN is at y = 2.75, length 6m. So half the chord length is 3m. So the chord MN goes from x = -3 to x = 3 at y = 2.75. Therefore, the endpoints of MN are M(-3, 2.75) and N(3, 2.75), and endpoints of PQ are P(-4, 0.75) and Q(4, 0.75).Now, the arc MP is from point M(-3, 2.75) to point P(-4, 0.75) along the semicircle. Wait, but this seems like moving from left to right? Wait, perhaps not. The arc could be the upper arc or the lower arc, but since it's a semicircle, the arcs are all the upper half. Wait, in the semicircle, the only arcs are the ones above the diameter. So the arc MP would be the part of the semicircle connecting M(-3, 2.75) to P(-4, 0.75). Wait, but these are two points on the semicircle. To find the arc length between them, we need the central angle between them.To compute the central angle, we can find the angle each point makes with the center O(0,0), then find the difference between those angles.Let me compute the coordinates of points M and P. Wait, as above:- Point M is (-3, 2.75)- Point P is (-4, 0.75)Wait, but hold on. Let's verify the coordinates. For chord MN at y = 2.75, length 6m. So half-length is 3m. So the x-coordinates are from -3 to +3. So MN is from (-3, 2.75) to (3, 2.75). Similarly, PQ is from (-4, 0.75) to (4, 0.75).But the problem mentions arc MP. If M is (-3, 2.75) and P is (-4, 0.75), then the arc from M to P would go clockwise around the semicircle. Alternatively, maybe the arc is from M to P through N and Q? Wait, but the problem just says arc MP. In a semicircle, the arc between two points is the shorter path along the circumference. Since it's a semicircle, the maximum arc length is πr. So the arc MP is either the path from M to P going one way or the other. But since both points are on the left side (negative x), the shorter arc would be going down from M to P. Wait, but in a semicircle, all arcs are less than or equal to πr. Let me compute the central angle between M and P.To compute the central angle, we can use the coordinates of M and P. The angle for each point can be found using the arctangent of y/x, but since the points are in the left half (negative x), the angles will be in the second quadrant.For point M(-3, 2.75):The angle θ₁ from the positive x-axis to point M is π - arctan(|y/x|) = π - arctan(2.75/3). Similarly, for point P(-4, 0.75):Angle θ₂ = π - arctan(0.75/4).Then, the central angle between M and P is θ₁ - θ₂.Compute θ₁ = π - arctan(2.75/3) ≈ π - arctan(0.9167). Let's compute arctan(0.9167). Since tan(42°) ≈ 0.9004, tan(43°) ≈ 0.9325, so 0.9167 is between 42° and 43°. Let's use a calculator: arctan(0.9167) ≈ 42.5°. Convert to radians: 42.5° * π/180 ≈ 0.7418 radians. So θ₁ ≈ π - 0.7418 ≈ 2.3998 radians.Similarly, for point P(-4, 0.75):θ₂ = π - arctan(0.75/4) = π - arctan(0.1875). arctan(0.1875) ≈ 10.6°, which is 0.185 radians. So θ₂ ≈ π - 0.185 ≈ 2.9565 radians.Therefore, the central angle between M and P is θ₂ - θ₁ ≈ 2.9565 - 2.3998 ≈ 0.5567 radians.Wait, but θ₂ is larger than θ₁ because point P is further to the left (more negative x) than point M? Wait, point P is at (-4, 0.75), which is further left (x = -4) than point M at (-3, 2.75). So in terms of angles from the positive x-axis, point P is at a larger angle (closer to π radians) than point M. Wait, no. Wait, the angle is measured from the positive x-axis going counterclockwise. Point M is at (-3, 2.75), which is in the second quadrant. The angle for M is π - arctan(2.75/3), and for P is π - arctan(0.75/4). Since 2.75/3 ≈ 0.9167 and 0.75/4 ≈ 0.1875, arctan(0.9167) ≈ 0.7418 radians, so θ₁ ≈ π - 0.7418 ≈ 2.3998 radians. arctan(0.1875) ≈ 0.185 radians, so θ₂ ≈ π - 0.185 ≈ 2.9565 radians. Therefore, the angle from M to P moving counterclockwise is θ₂ - θ₁ ≈ 0.5567 radians. Alternatively, going clockwise, it would be 2π - (θ₂ - θ₁), but since it's a semicircle, the maximum central angle is π. So the smaller arc is θ₂ - θ₁ ≈ 0.5567 radians.But wait, in the semicircle, the total circumference is πr. The arc length MP is r * Δθ, where Δθ is the central angle in radians. So if Δθ ≈ 0.5567 radians, then arc length is ≈ 4.0697 * 0.5567 ≈ 2.265 meters. But let's check this more accurately.Alternatively, maybe there's a better way to compute the central angle. Since we have coordinates of points M and P, we can use the dot product formula. The central angle between two points can be found using the dot product of their position vectors.For point M(-3, 2.75) and point P(-4, 0.75):The vectors OM and OP are (-3, 2.75) and (-4, 0.75), respectively.The dot product is (-3)(-4) + (2.75)(0.75) = 12 + 2.0625 = 14.0625.The magnitudes of OM and OP are both equal to the radius r ≈ 4.0697 m. Wait, but let's compute them exactly:|OM| = sqrt((-3)^2 + (2.75)^2) = sqrt(9 + 7.5625) = sqrt(16.5625) = 4.0697 m, which matches the radius.Similarly, |OP| = sqrt((-4)^2 + (0.75)^2) = sqrt(16 + 0.5625) = sqrt(16.5625) = 4.0697 m. Correct.So the dot product formula:OM · OP = |OM||OP|cosθ ⇒ cosθ = (OM · OP)/(|OM||OP|) = 14.0625 / (4.0697^2)Compute 4.0697^2 ≈ 16.5625Therefore, cosθ ≈ 14.0625 / 16.5625 ≈ 0.8495.Then θ = arccos(0.8495) ≈ 31.7 degrees, which is approximately 0.553 radians.Wait, but earlier calculation gave 0.5567 radians. These are close, slight difference due to approximate calculations. So θ ≈ 0.553 radians.Therefore, the arc length MP is rθ ≈ 4.0697 * 0.553 ≈ 2.252 meters.But let's do this more accurately.First, compute OM · OP:(-3)(-4) + (2.75)(0.75) = 12 + 2.0625 = 14.0625|OM| = |OP| = sqrt(16.5625) = 4.0625 m. Wait, wait, hold on. Earlier, we had r² = 16.5625, so r = sqrt(16.5625) = 4.0625 m. Wait, 4.0625 squared is (4 + 0.0625)^2 = 16 + 0.5 + 0.00390625 = 16.50390625. Wait, but 16.5625 is 4.0625 squared? Wait, 4.0625 * 4.0625 = ?4 * 4 = 164 * 0.0625 = 0.250.0625 * 4 = 0.250.0625 * 0.0625 = 0.00390625So total is 16 + 0.25 + 0.25 + 0.00390625 = 16.50390625. Wait, but we have r² = 16.5625. Therefore, sqrt(16.5625) is exactly 4.0625?Wait, 4.0625 * 4.0625:4 * 4 = 164 * 0.0625 = 0.250.0625 * 4 = 0.250.0625 * 0.0625 = 0.00390625Adding them up: 16 + 0.25 + 0.25 + 0.00390625 = 16.50390625. So this is not 16.5625. Wait, so there must be a miscalculation. Wait, earlier, in equation (1):r² = 16 + h² = 16 + (0.75)^2 = 16 + 0.5625 = 16.5625. So r² is 16.5625, so r = sqrt(16.5625). But 4.0625 squared is 16.50390625, which is less than 16.5625. So sqrt(16.5625) is not 4.0625. Then what is it?Wait, 16.5625 divided by 16.50390625 is approximately 1.0035. So sqrt(16.5625) ≈ 4.0625 * sqrt(1.0035) ≈ 4.0625 * 1.00175 ≈ 4.0697, as before.But in reality, 16.5625 is equal to 265/16, since 16*16=256, 256 +9=265, so 265/16=16.5625. So sqrt(265/16)=sqrt(265)/4. sqrt(265) is approximately 16.2788. Therefore, sqrt(265)/4 ≈ 4.0697.So the exact value is sqrt(265)/4. So when computing the dot product, OM · OP = 14.0625.Therefore, cosθ = 14.0625 / ( (sqrt(265)/4)^2 ) = 14.0625 / (265/16) ) = 14.0625 * (16/265) = (14.0625 * 16)/265.Calculate numerator: 14.0625 * 16. 14*16=224, 0.0625*16=1, so total 225. Therefore, cosθ = 225 / 265 ≈ 0.8490566.Therefore, θ = arccos(0.8490566). Let's compute this in radians.arccos(0.8490566) ≈ 0.556 radians. Let's verify:cos(0.556) ≈ cos(31.8 degrees). Let's convert 0.556 radians to degrees: 0.556 * (180/π) ≈ 31.86 degrees. cos(31.86 degrees) ≈ 0.849, which matches. So θ ≈ 0.556 radians.Therefore, the arc length MP is r * θ ≈ (sqrt(265)/4) * 0.556. Let's compute sqrt(265) ≈ 16.2788, so sqrt(265)/4 ≈ 4.0697. Multiply by 0.556: 4.0697 * 0.556 ≈ 2.262 meters.But let's compute this more accurately. Let's use exact expressions where possible.Given that r = sqrt(265)/4, and θ = arccos(225/265).So arc length MP = r * θ = (sqrt(265)/4) * arccos(225/265).We can compute this numerically.First, compute 225/265 ≈ 0.8490566.Compute arccos(0.8490566):Using a calculator, arccos(0.8490566) ≈ 31.85 degrees. Convert to radians: 31.85 * π/180 ≈ 0.5558 radians.Then, multiply by r ≈ 4.0697:4.0697 * 0.5558 ≈ 2.262 meters.But let's check using the exact value of r and more precise angle.Alternatively, we can use the chord length formula. The chord length between M and P is the distance between points M(-3, 2.75) and P(-4, 0.75).Distance MP = sqrt[ (-4 + 3)^2 + (0.75 - 2.75)^2 ] = sqrt[ (-1)^2 + (-2)^2 ] = sqrt[1 + 4] = sqrt(5) ≈ 2.23607 meters.But chord length MP is related to the central angle by the formula:Chord length = 2r sin(θ/2)So we have:sqrt(5) = 2 * (sqrt(265)/4) * sin(θ/2)Simplify:sqrt(5) = (sqrt(265)/2) * sin(θ/2)Solve for sin(θ/2):sin(θ/2) = (2 sqrt(5)) / sqrt(265) = 2 sqrt(5) / sqrt(265) = 2 sqrt(5/265) = 2 sqrt(1/53) ≈ 2 * 0.1374 ≈ 0.2748Then θ/2 = arcsin(0.2748) ≈ 0.278 radians, so θ ≈ 0.556 radians, which matches previous results. Therefore, θ ≈ 0.556 radians.Therefore, arc length MP ≈ r * θ ≈ 4.0697 * 0.556 ≈ 2.262 meters.But we need to present the answer accurate to 0.001 meters. So we need to compute this more precisely.First, compute θ = arccos(225/265). Let's compute 225/265 = 45/53 ≈ 0.8490566037735849.Compute arccos(45/53):Using a calculator, arccos(45/53) ≈ 0.5558 radians.Then, multiply by r = sqrt(265)/4 ≈ 16.2788/4 ≈ 4.0697.So 4.0697 * 0.5558 ≈ let's compute this:4 * 0.5558 = 2.22320.0697 * 0.5558 ≈ 0.0387Total ≈ 2.2232 + 0.0387 ≈ 2.2619 meters.Rounded to three decimal places: 2.262 meters.But let's check using higher precision calculations.Alternatively, use exact expressions:θ = arccos(45/53)r = sqrt(265)/4So arc length = (sqrt(265)/4) * arccos(45/53)Compute sqrt(265):sqrt(265) ≈ 16.278820596099706arccos(45/53):Using a calculator, arccos(45/53) in radians ≈ 0.5557041453664163Multiply: 16.278820596099706 / 4 * 0.5557041453664163 ≈ 4.069705149024927 * 0.5557041453664163 ≈ 2.2619 meters.So approximately 2.262 meters.But the problem asks for the length of arc MP accurate to 0.001 meters. So the answer should be 2.262 meters. However, let's confirm once more.Alternatively, using coordinates, we can parametrize the points on the semicircle and compute the angle.Points M(-3, 2.75) and P(-4, 0.75) lie on the circle x² + y² = (sqrt(16.5625))² = 16.5625.The angle for each point can be calculated as follows:For point M(-3, 2.75):tanθ₁ = y/x = 2.75 / (-3) = -0.916666...But since the point is in the second quadrant, θ₁ = π - arctan(0.916666...) ≈ π - 0.7417 ≈ 2.3998 radians.For point P(-4, 0.75):tanθ₂ = y/x = 0.75 / (-4) = -0.1875θ₂ = π - arctan(0.1875) ≈ π - 0.1853 ≈ 2.9563 radians.Central angle between M and P is θ₂ - θ₁ ≈ 2.9563 - 2.3998 ≈ 0.5565 radians.Multiply by radius r ≈ 4.0697:0.5565 * 4.0697 ≈ 0.5565 * 4 + 0.5565 * 0.0697 ≈ 2.226 + 0.0388 ≈ 2.2648 meters.Hmm, slight differences due to rounding in intermediate steps. So with more precise calculation, it's approximately 2.2648 meters. Rounded to three decimal places, 2.265 meters.But the previous calculation was 2.2619 meters. There's a discrepancy here due to the precision in intermediate steps. Let's use more precise values.Compute θ₁:For point M(-3, 2.75):tanθ = |2.75 / 3| = 0.916666...arctan(0.916666...) = arctan(11/12). Using a calculator, arctan(11/12) ≈ 0.7417557301 radians.So θ₁ = π - 0.7417557301 ≈ 3.1415926536 - 0.7417557301 ≈ 2.3998369235 radians.For point P(-4, 0.75):tanθ = |0.75 / 4| = 0.1875arctan(0.1875) ≈ 0.1853479495 radians.θ₂ = π - 0.1853479495 ≈ 3.1415926536 - 0.1853479495 ≈ 2.9562447041 radians.Central angle Δθ = θ₂ - θ₁ ≈ 2.9562447041 - 2.3998369235 ≈ 0.5564077806 radians.Multiply by radius r = sqrt(265)/4 ≈ 16.2788205961 / 4 ≈ 4.0697051490.So arc length MP ≈ 4.0697051490 * 0.5564077806 ≈ Let's compute this:First, 4 * 0.5564077806 = 2.22563112240.0697051490 * 0.5564077806 ≈ 0.069705149 * 0.5 = 0.0348525745, 0.069705149 * 0.0564077806 ≈ 0.003938. Total ≈ 0.0348525745 + 0.003938 ≈ 0.0387905745Total arc length ≈ 2.2256311224 + 0.0387905745 ≈ 2.2644216969 meters.Rounded to three decimal places, 2.264 meters.However, different methods give slightly different results due to rounding at different steps. To ensure accuracy, perhaps use more precise intermediate values or exact formulas.Alternatively, use the chord length formula:Chord length MP = sqrt(5) ≈ 2.2360679775 meters.Chord length = 2r sin(θ/2) ⇒ 2.2360679775 = 2 * 4.0697051490 * sin(θ/2)Solve for sin(θ/2):sin(θ/2) = 2.2360679775 / (2 * 4.0697051490) ≈ 2.2360679775 / 8.139410298 ≈ 0.2747211275.θ/2 = arcsin(0.2747211275) ≈ 0.27825 radians.Therefore, θ ≈ 0.5565 radians.Then arc length = rθ ≈ 4.0697051490 * 0.5565 ≈ 2.2644 meters.So, consistently, it's approximately 2.264 meters when calculated precisely. However, the problem asks for the answer accurate to 0.001 meters, so 2.264 meters.But wait, let's check with the exact formula.We have:θ = arccos(45/53)r = sqrt(265)/4Compute θ in radians to high precision:Using a calculator, arccos(45/53) ≈ 0.5557041453664163 radians.Multiply by r ≈ 4.069705149024927:0.5557041453664163 * 4.069705149024927 ≈ 2.2619308579 meters.Wait, this is conflicting with previous results. Wait, why is there inconsistency?Because when using the chord length method, we get θ ≈ 0.5565, but when using the dot product method, we get θ ≈ 0.5557. The difference is due to rounding in the chord length calculation. The chord length MP is exactly sqrt(5) ≈ 2.2360679775, which is exact. Therefore, using chord length:sin(θ/2) = sqrt(5)/(2r) = sqrt(5)/(2*sqrt(265)/4) = (sqrt(5)*4)/(2*sqrt(265)) = (2 sqrt(5))/sqrt(265) = 2*sqrt(5/265) = 2*sqrt(1/53) ≈ 2*0.13736056 ≈ 0.27472112Then θ/2 = arcsin(0.27472112) ≈ 0.27824967 radiansTherefore, θ ≈ 0.55649934 radiansMultiply by r = sqrt(265)/4 ≈ 4.0697051490.55649934 * 4.069705149 ≈ let's compute this more accurately:0.5 * 4.069705149 = 2.03485257450.05649934 * 4.069705149 ≈ 0.05649934 * 4 = 0.22599736, 0.05649934 * 0.069705149 ≈ 0.0039439. Total ≈ 0.22599736 + 0.0039439 ≈ 0.22994126Total ≈ 2.0348525745 + 0.22994126 ≈ 2.26479383 meters.So approximately 2.264794 meters, which rounds to 2.265 meters.However, using the dot product formula gives θ ≈ 0.5557 radians, leading to arc length ≈ 2.2619 meters. The discrepancy arises because the chord length formula uses the exact chord length, while the dot product formula might have used approximate coordinates.Wait, let's check the coordinates. Wait, chord PQ is length 8m at y = 0.75. So half-length is 4m, so x = ±4. But in the circle equation x² + y² = r², at y = 0.75, x² = r² - 0.75² = 16.5625 - 0.5625 = 16. Therefore, x = ±4. So PQ endpoints are (-4, 0.75) and (4, 0.75). Correct.Similarly, chord MN at y = 2.75, length 6m. Half-length 3m, so x = ±3. Check x² + y² = (-3)^2 + 2.75^2 = 9 + 7.5625 = 16.5625 = r². Correct. So coordinates are accurate.Therefore, the chord length MP is indeed sqrt( [(-4 + 3)^2 + (0.75 - 2.75)^2] ) = sqrt(1 + 4) = sqrt(5). Therefore, chord length is exact.Using the chord length formula gives a more accurate result because the chord length is exact, while the dot product calculation might have used approximate angles. Therefore, relying on the chord length formula:θ ≈ 0.5565 radians, leading to arc length ≈ 2.2648 meters, which is 2.265 meters when rounded to three decimal places.However, the answer obtained through the precise chord length method is approximately 2.265 meters, while the dot product method gave 2.262 meters. This inconsistency is concerning. I need to resolve this.Alternatively, perhaps the problem is expecting a different approach. Let's think again.We have two parallel chords MN and PQ in a semicircle, distance between them is 2m, lengths 6m and 8m. We found the radius r ≈ 4.0697m. Now, to find the length of arc MP.Assuming that points M and P are on the same side of the vertical diameter (i.e., both on the left side or both on the right side), then the arc MP would be the arc between them along the semicircle. However, depending on their positions, the arc could span a certain angle.Alternatively, perhaps there's a formula relating the arc length between two parallel chords. However, I don't recall such a formula. Alternatively, since we know the radius and the coordinates, the accurate method is to compute the central angle.Given the accurate chord length method gives us θ ≈ 0.5565 radians, arc length ≈ 2.2648 meters ≈ 2.265 meters.However, let's verify using parametric angles.Alternatively, compute the angle for each point using coordinates.For point M(-3, 2.75):This is in the second quadrant. The angle θ₁ = π - arctan(2.75/3). Compute arctan(2.75/3):2.75/3 = 0.916666...arctan(0.916666...) ≈ 0.7417557301 radians.Thus, θ₁ ≈ π - 0.7417557301 ≈ 2.3998369235 radians.For point P(-4, 0.75):This is also in the second quadrant. Angle θ₂ = π - arctan(0.75/4).0.75/4 = 0.1875arctan(0.1875) ≈ 0.1853479495 radians.Thus, θ₂ ≈ π - 0.1853479495 ≈ 2.9562447041 radians.Central angle Δθ = θ₂ - θ₁ ≈ 2.9562447041 - 2.3998369235 ≈ 0.5564077806 radians.Multiply by radius r ≈ 4.0697051490:0.5564077806 * 4.0697051490 ≈ 2.2644216969 meters ≈ 2.264 meters.However, when using the chord length method, we have:Chord length MP = sqrt(5) ≈ 2.2360679775 meters.Using chord length formula: chord length = 2r sin(θ/2). Therefore:2.2360679775 = 2 * 4.0697051490 * sin(θ/2)sin(θ/2) = 2.2360679775 / (8.139410298) ≈ 0.2747211275.θ/2 = arcsin(0.2747211275) ≈ 0.2782496735 radians.θ ≈ 0.5564993470 radians.Multiply by r ≈ 4.0697051490:0.5564993470 * 4.0697051490 ≈ 2.2647938341 meters ≈ 2.265 meters.Hence, both methods give approximately 2.265 meters when calculated precisely. The difference between 2.264 and 2.265 is due to rounding at different steps. Given the problem asks for accuracy to 0.001 meters, the answer should be 2.265 meters.However, let's cross-verify with the initial approximate calculation using the dot product, which gave 2.262 meters. The slight discrepancy arises from the precision in intermediate steps. Given that the chord length method and the angle difference method both give around 2.2648-2.265 meters when calculated with higher precision, I would conclude that the accurate answer is 2.265 meters.But to ensure correctness, perhaps we should use more exact computations.Given that:θ = arccos( (OM · OP) / r² ) = arccos(14.0625 / 16.5625) = arccos(225/265) = arccos(45/53).Using a calculator to compute arccos(45/53):45/53 ≈ 0.8490566038arccos(0.8490566038) ≈ 0.5557041454 radians.Multiply by r = sqrt(265)/4 ≈ 4.0697051490:0.5557041454 * 4.0697051490 ≈ 2.2619308575 meters ≈ 2.262 meters.But this conflicts with the previous results. So now there's a problem. Three different methods give slightly different results:1. Angle difference: 2.264 m2. Chord length formula: 2.265 m3. Dot product formula: 2.262 mThis inconsistency is due to rounding errors in intermediate steps. To resolve this, let's use higher precision calculations.First, compute θ using the chord length formula, which is more reliable as chord length is exact:Chord length MP = sqrt(5) = 2.2360679775Using chord length formula:Chord length = 2r sin(θ/2)=> sin(θ/2) = sqrt(5)/(2r) = sqrt(5)/(2*(sqrt(265)/4)) = (4 sqrt(5))/(2 sqrt(265)) = (2 sqrt(5))/sqrt(265)Rationalizing the denominator:= 2 sqrt(5) / sqrt(265) = 2 sqrt(5 * 265) / 265 = 2 sqrt(1325) / 265 ≈ 2 * 36.40054945 / 265 ≈ 72.8010989 / 265 ≈ 0.2747211275Compute arcsin(0.2747211275) in radians to high precision:Using a calculator, arcsin(0.2747211275) ≈ 0.2782496735 radians.Thus, θ ≈ 0.5564993470 radians.Multiply by r = sqrt(265)/4 ≈ 4.0697051490:0.5564993470 * 4.0697051490 ≈0.5 * 4.0697051490 = 2.03485257450.0564993470 * 4.0697051490 ≈ 0.0564993470 * 4 = 0.225997388; 0.0564993470 * 0.0697051490 ≈ 0.0039439. Total ≈ 0.225997388 + 0.0039439 ≈ 0.229941288.Total θ*r ≈ 2.0348525745 + 0.229941288 ≈ 2.2647938625 meters ≈ 2.264794 meters.Rounded to three decimal places: 2.265 meters.However, the precise value is approximately 2.264794 meters, which is 2.265 when rounded to three decimal places.Alternatively, using high-precision calculation for arccos(45/53):Using a calculator, arccos(45/53) ≈ 0.5557041453664163 radians.Multiply by r = sqrt(265)/4 ≈ 4.069705149024927:0.5557041453664163 * 4.069705149024927 ≈ 2.261930857949511 meters ≈ 2.262 meters.This discrepancy suggests that there is an inconsistency between the chord length method and the dot product method. This is confusing.Wait, but perhaps I made a mistake in assuming which points are connected by the arc MP. If the arc is supposed to be the other arc (the major arc instead of the minor arc), but in a semicircle, the maximum arc length is πr ≈ 12.789 meters, which is way larger. But the calculated arc length is around 2.26 meters, which is plausible as a minor arc.Alternatively, perhaps the problem is not referring to the arc between points M and P as I assumed, but another arc.Wait, in the problem statement, it says "the length of overparen{MP}". The term "overparen" suggests the arc from M to P passing through N and Q, i.e., the major arc. But in a semicircle, the major arc would be the same as the minor arc because it's only half a circle. Wait, no. In a full circle, you have major and minor arcs, but in a semicircle, any arc between two points is either the direct arc or the reflex arc, but the reflex arc would go the other way around the semicircle. However, since it's a semicircle, the reflex arc would actually coincide with the other arc but in the opposite direction. However, the length would be the same because it's a semicircle. Wait, no. In a semicircle, any two points divide the semicircle into two arcs: one smaller and one larger. However, since it's a semicircle, the larger arc can't exceed πr. For example, if two points are very close, the minor arc is small, and the major arc is the rest up to πr.But in our case, points M and P are on the semicircle. The central angle between them is about 0.556 radians, so the major arc would be πr - 0.556r ≈ 3.1416*4.0697 - 0.556*4.0697 ≈ 12.789 - 2.264 ≈ 10.525 meters. But the problem asks for "the length of overparen{MP}", which is most likely the minor arc, as it's the shorter one.But in Chinese mathematical problems, "overparen" (弧) typically refers to the minor arc unless specified otherwise. Therefore, the answer should be approximately 2.265 meters.However, the precise calculation using the chord length method gives 2.264794, which is 2.265 when rounded to three decimal places. The dot product method gives 2.2619308579, which is 2.262. This discrepancy is due to different methods and rounding. To resolve this, perhaps the exact value is 2.265 meters.Alternatively, since the problem might expect an answer derived from the central angle found through geometric relations rather than coordinate geometry.Wait, another approach: the positions of the chords can be related through the radius and their distances from the center. Once we have the radius, we can find the angles corresponding to each chord, then find the difference.For chord PQ at distance h = 0.75 from the center:The angle subtended by PQ at the center is 2 arcsin( (PQ/2) / r ) = 2 arcsin(4 / r). Similarly, for chord MN at distance h + 2 = 2.75 from the center, the angle is 2 arcsin(3 / r).But wait, this is the angle subtended by each chord. However, we need the angle between the two chords.Wait, if we consider the angles from the center to each chord, the angle between the two chords would be the difference between their individual angles.Wait, actually, each chord is horizontal, so their endpoints are symmetrically placed around the vertical diameter. The angle for each chord can be measured from the vertical diameter.Alternatively, since the chords are horizontal, the angle between the radii to their endpoints can be computed.For chord PQ at y = 0.75, length 8m, radius r ≈ 4.0697. The angle subtended by PQ at the center is 2θ, where θ = arcsin(4 / r). So θ = arcsin(4 / 4.0697) ≈ arcsin(0.983) ≈ 1.387 radians. Therefore, the angle subtended by PQ is 2 * 1.387 ≈ 2.774 radians.Similarly, for chord MN at y = 2.75, length 6m, the angle subtended is 2φ, where φ = arcsin(3 / r) ≈ arcsin(3 / 4.0697) ≈ arcsin(0.737) ≈ 0.827 radians. Therefore, angle subtended by MN is 2 * 0.827 ≈ 1.654 radians.But how does this help in finding the angle between MP?Alternatively, if we consider the positions of M and P, which are endpoints of the chords, the angle between them can be found by considering the angular difference between their positions.If we consider the rightmost point of PQ as Q(4, 0.75) and the rightmost point of MN as N(3, 2.75), but the arc in question is MP, which are both left endpoints: M(-3, 2.75) and P(-4, 0.75).The central angle between M and P is the difference in their angles from the center. As previously calculated, this is approximately 0.5565 radians, leading to an arc length of approximately 2.265 meters.Given that all methods when calculated precisely converge to around 2.265 meters, considering rounding to three decimal places, the answer should be 2.265 meters.However, to cross-validate, let's use approximate values:Given r ≈ 4.0697 m, central angle ≈ 0.5565 radians.Arc length ≈ 4.0697 * 0.5565 ≈ 4.0697 * 0.5 + 4.0697 * 0.0565 ≈ 2.03485 + 0.2298 ≈ 2.2646 ≈ 2.265 m.Therefore, the length of arc MP is approximately 2.265 meters.Final Answerboxed{2.265}</think>Given a semicircle with center ( O ) and two parallel chords ( MN ) and ( PQ ) such that ( MN = 6 , text{m} ), ( PQ = 8 , text{m} ), and the distance between them is ( 2 , text{m} ), we need to find the length of the arc ( overparen{MP} ).1. Determine the radius of the semicircle: - Let ( h ) be the distance from the center ( O ) to chord ( PQ ). The distance from ( O ) to chord ( MN ) is ( h + 2 ). - Using the chord length formula ( l = 2sqrt{r^2 - d^2} ): - For ( PQ ): ( 8 = 2sqrt{r^2 - h^2} ) ⟹ ( sqrt{r^2 - h^2} = 4 ) ⟹ ( r^2 - h^2 = 16 ). - For ( MN ): ( 6 = 2sqrt{r^2 - (h + 2)^2} ) ⟹ ( sqrt{r^2 - (h + 2)^2} = 3 ) ⟹ ( r^2 - (h + 2)^2 = 9 ). - Subtracting these equations: ( (r^2 - h^2) - (r^2 - (h + 2)^2) = 16 - 9 ) ⟹ ( 4h + 4 = 7 ) ⟹ ( h = 0.75 , text{m} ). - Substituting ( h = 0.75 ) into ( r^2 - h^2 = 16 ): ( r^2 = 16.5625 ) ⟹ ( r = sqrt{16.5625} approx 4.0697 , text{m} ).2. Calculate the central angle ( theta ) for arc ( MP ): - Coordinates of points ( M(-3, 2.75) ) and ( P(-4, 0.75) ). - Using the dot product to find the central angle: - Vectors ( overrightarrow{OM} = (-3, 2.75) ) and ( overrightarrow{OP} = (-4, 0.75) ). - Dot product: ( overrightarrow{OM} cdot overrightarrow{OP} = 14.0625 ). - Magnitudes: ( |overrightarrow{OM}| = |overrightarrow{OP}| = sqrt{16.5625} approx 4.0697 ). - ( cos theta = frac{14.0625}{16.5625} approx 0.8490566 ) ⟹ ( theta approx arccos(0.8490566) approx 0.5557 , text{radians} ).3. Calculate the arc length ( overparen{MP} ): - Arc length ( = r theta approx 4.0697 times 0.5557 approx 2.2619 , text{m} ). - Using the chord length method for verification: - Chord length ( MP = sqrt{5} approx 2.23607 , text{m} ). - ( sin(theta/2) = frac{sqrt{5}}{2r} approx 0.2747 ) ⟹ ( theta approx 0.5565 , text{radians} ). - Arc length ( approx 4.0697 times 0.5565 approx 2.2648 , text{m} ).After precise calculations and rounding to three decimal places, the length of arc ( overparen{MP} ) is:[boxed{2.265}],<think>Okay, let's try to solve this problem. So, we have a semicircle with center O. The shaded part is a closed figure within the semicircle. There are two chords, MN and PQ, which are parallel. The distance between them is 2 meters. MN is 6 meters long, and PQ is 8 meters long. We need to find the length of the arc MP, accurate to 0.001 meters.First, let me visualize the semicircle. Since it's a semicircle, the diameter is the base, and the center O is at the midpoint of this diameter. The chords MN and PQ are parallel, and they are inside the semicircle. The distance between MN and PQ is 2 meters. MN is shorter than PQ, so I think MN must be closer to the top of the semicircle (the center O), and PQ is closer to the diameter. Because usually, in a circle, longer chords are closer to the center. But since this is a semicircle, maybe the diameter is the base, and the chords are above the diameter?Wait, actually, in a full circle, longer chords are closer to the center. In a semicircle, which is half of a circle, the same principle should apply. So, if PQ is longer (8m) than MN (6m), then PQ is closer to the center O, and MN is higher up, closer to the arc of the semicircle. The distance between them is 2 meters. So, MN is 2 meters above PQ.Wait, but since the semicircle is only half, maybe the center O is on the diameter, which is the base of the semicircle. So, the semicircle is drawn with diameter, let's say, AB, and O is the center. Then, the chords MN and PQ are parallel to each other and to the diameter AB? Wait, the problem says "chord MN is parallel to chord PQ", but it doesn't say they are parallel to the diameter. Hmm.But maybe they are not horizontal? Wait, but in a semicircle, the diameter is the base, and chords can be at different heights. If they are parallel, they must be horizontal? Wait, no. Wait, in a circle, chords that are parallel can be at any angle, but in a semicircle, if it's drawn with the diameter horizontal, then parallel chords would be horizontal as well. But maybe the problem assumes that the semicircle is oriented such that the diameter is horizontal, so the chords MN and PQ are horizontal lines. So, their distance apart is vertical distance 2 meters.So, MN is 6 meters long, PQ is 8 meters long, separated by 2 meters vertically. The figure is a closed region between these two chords and the semicircle. We need to find the length of the arc MP.Wait, the arc MP. So, points M and P are the endpoints of the arc. Since MN and PQ are chords, maybe M is one end of chord MN, and P is one end of chord PQ. If the chords are parallel, then the arc MP would be part of the semicircle's circumference.But to find the length of the arc MP, we need the angle subtended by MP at the center O, and then compute the arc length as rθ, where θ is in radians.So, first, we need to determine the radius of the semicircle. Wait, the problem doesn't specify the radius. So we need to find the radius first, using the given chords and their distances.Let me try to model this. Let's consider the semicircle with center O, diameter AB. Let’s place the center O at the origin (0,0) for coordinate system purposes, with AB along the x-axis from (-r,0) to (r,0), and the semicircle above the x-axis.Chords MN and PQ are horizontal lines (since they are parallel and the distance between them is vertical). MN is higher up, with length 6m, PQ is lower, with length 8m, and the vertical distance between them is 2m.Let’s denote the distance from the center O to chord PQ as h. Then, since PQ is 8m long, we can use the formula for the length of a chord: length = 2√(r² - d²), where d is the distance from the center to the chord.Similarly, the distance from center O to chord MN is h + 2 (since the vertical distance between MN and PQ is 2m). The length of MN is 6m, so 6 = 2√(r² - (h + 2)²), and length of PQ is 8 = 2√(r² - h²).So, we can set up two equations:For PQ: 8 = 2√(r² - h²) ⇒ 4 = √(r² - h²) ⇒ 16 = r² - h² ⇒ r² = h² + 16. (1)For MN: 6 = 2√(r² - (h + 2)^2) ⇒ 3 = √(r² - (h² + 4h + 4)) ⇒ 9 = r² - h² - 4h - 4 ⇒ r² = h² + 4h + 13. (2)From equation (1): r² = h² + 16From equation (2): r² = h² + 4h + 13Set equal: h² + 16 = h² + 4h + 13 ⇒ 16 = 4h + 13 ⇒ 4h = 3 ⇒ h = 3/4 = 0.75 meters.Then, substituting h = 0.75 into equation (1): r² = (0.75)^2 + 16 = 0.5625 + 16 = 16.5625 ⇒ r = √16.5625 = 4.0625 meters. Wait, √16.5625. Let's calculate that:16.5625 = 16 + 9/16 = (16*16 + 9)/16 = (256 + 9)/16 = 265/16. Hmm, but 16.5625 is 4.0625², since 4² = 16, 0.0625² ≈ 0.00390625, but wait, (4 + 0.0625)² = 16 + 2*4*0.0625 + 0.0625² = 16 + 0.5 + 0.00390625 = 16.50390625. Wait, that's not matching. Wait, perhaps better to compute 16.5625.But 4.0625 squared is 4.0625 * 4.0625. Let me compute:4 * 4 = 164 * 0.0625 = 0.250.0625 * 4 = 0.250.0625 * 0.0625 = 0.00390625So, (4 + 0.0625)^2 = 16 + 2*(4*0.0625) + (0.0625)^2 = 16 + 0.5 + 0.00390625 = 16.50390625. Hmm, but 16.5625 is larger. Wait, maybe another approach. Wait, 4.1^2 = 16.81, which is more than 16.5625, so 4.0625^2 is 16.5039, which is less than 16.5625. Wait, maybe my calculation is wrong. Wait, 16.5625. Let me see, 16.5625 is 16 + 0.5625. 0.5625 is 9/16. So sqrt(16 + 9/16). Hmm, sqrt(16.5625). Let's check 4.06^2: 4.06*4.06 = 16.4836. 4.07^2=16.5649. Ah! So, 4.07^2=16.5649, which is very close to 16.5625. So, sqrt(16.5625) is 4.07 approximately. Wait, but 4.07^2 is 16.5649, which is slightly more than 16.5625, so the exact value is 4.07 - a tiny bit. Wait, but maybe 16.5625 is a perfect square? Let me check.16.5625 * 10000 = 165625. What's the square root of 165625? 165625 divided by 25 is 6625. 6625 divided by 25 is 265. So sqrt(165625) = 25*sqrt(265). Hmm, maybe not. Wait, 407 * 407 = 165,649. So 407^2 = 165,649, which is 165625 + 24. So, 407^2 is 165,649. So 165,625 is 165,649 -24. So, sqrt(165,625) ≈ 407 - 24/(2*407) ≈ 407 - 12/407 ≈ 407 - 0.0295 ≈ 406.9705. Therefore, sqrt(16.5625) = sqrt(165625/10000) = 406.9705/100 ≈ 4.069705. So approximately 4.0697 meters.But maybe the exact value is 4.0625? Wait, 4.0625 is 4 and 1/16, which is 65/16. (65/16)^2 = (4225)/256 ≈ 16.50390625, which is not 16.5625. So, maybe the problem expects us to keep it in fractional form? Wait, perhaps not. Let's check.Wait, we had h = 0.75 m. So, substituting h = 0.75 into equation (1):r² = (0.75)^2 + 16 = 0.5625 + 16 = 16.5625. So, r is sqrt(16.5625). But as we saw, sqrt(16.5625) is 4.0625? Wait, no, wait. Wait, 4.0625 squared is 16.50390625, which is not 16.5625. Wait, maybe I miscalculated h?Wait, earlier steps:From the two equations:From PQ: 8 = 2√(r² - h²) ⇒ √(r² - h²) = 4 ⇒ r² - h² = 16 (1)From MN: 6 = 2√(r² - (h + 2)^2) ⇒ √(r² - (h + 2)^2) = 3 ⇒ r² - (h² + 4h + 4) = 9 ⇒ r² - h² -4h -4 = 9 ⇒ r² - h² = 4h + 13 (2)From (1): r² - h² = 16From (2): r² - h² = 4h + 13So, 16 = 4h + 13 ⇒ 4h = 3 ⇒ h = 3/4 = 0.75. That's correct.Then, r² = h² + 16 = (0.75)^2 + 16 = 0.5625 + 16 = 16.5625. So, r = sqrt(16.5625). As we saw, sqrt(16.5625) is 4.07 meters approximately. But maybe exact value is 4.07? Wait, 4.07^2 = 16.5649, which is very close. So, r ≈ 4.07 meters. But we can also write r as 4.0625? Wait, 4.0625 is 4 + 1/16, but 1/16 is 0.0625. Wait, 4.0625 squared is 16.50390625, which is not 16.5625. So, actually, the exact value is √(16.5625). Let me see. 16.5625 can be written as 16 + 0.5625. 0.5625 is 9/16. So sqrt(16 + 9/16). Hmm, perhaps there's a better way. Wait, 16.5625 * 16 = 265. So sqrt(265/16) = sqrt(265)/4. Since 265 is 5*53, which is not a perfect square. So, we can keep it as sqrt(265)/4. But sqrt(265) is approximately 16.2788, so sqrt(265)/4 ≈ 4.0697. So, r ≈ 4.0697 meters.Okay, so the radius is approximately 4.07 meters.Now, we need to find the length of the arc MP. For that, we need the angle between points M and P as viewed from the center O.Assuming that M is an endpoint of chord MN and P is an endpoint of chord PQ. Since the chords are parallel and horizontal, the points M and P would be on the same vertical line? Wait, no. If MN and PQ are horizontal chords, then their endpoints are symmetric with respect to the vertical diameter. So, if MN is a horizontal chord above PQ, then the left endpoint of MN is M, and the left endpoint of PQ is P. Then, the arc MP would go from M to P along the semicircle.But actually, since the semicircle is from A to B, the diameter. If MN and PQ are horizontal chords above the diameter AB, then points M and N are on MN, and points P and Q are on PQ. Since MN and PQ are horizontal, their endpoints are symmetric around the vertical diameter. So, for example, point M is on the left side of chord MN, point N is on the right. Similarly, point P is on the left of chord PQ, point Q is on the right.Therefore, the arc MP goes from point M (left end of upper chord MN) to point P (left end of lower chord PQ). To find the length of this arc, we need the central angle between M and P.To find that angle, we can find the coordinates of points M and P, compute the angle each makes with the center, and subtract them.First, let's set up the coordinate system. Let’s take center O at (0,0). The semicircle is the upper half of the circle with radius r ≈ 4.0697 meters.Chord PQ is at distance h = 0.75 meters from the center. Since it's a horizontal chord, its equation is y = h = 0.75 m. The chord PQ has length 8 meters, so half-length is 4 meters. Therefore, the endpoints P and Q are at ( -4, 0.75 ) and ( 4, 0.75 ), but wait, no. Wait, the chord is at y = h = 0.75, and its length is 8 meters. The horizontal distance from the center to each endpoint is sqrt(r² - h²). Wait, but we already used that. Wait, the chord length is 8 meters, so half the chord length is 4 meters. Therefore, the horizontal distance from the center (along the x-axis) to each endpoint is 4 meters. Wait, but that can't be, because if the chord is horizontal at y = h, then the horizontal distance from the center to the chord is not 4 meters. Wait, no, the chord is of length 8, so half-length is 4. The horizontal distance from the center to each endpoint is sqrt(r² - h²). Wait, no.Wait, the chord PQ is horizontal at y = h = 0.75. The endpoints of PQ are located at (x, 0.75) where x satisfies x² + (0.75)^2 = r². Wait, no. Wait, the chord is part of the semicircle, so the endpoints are on the semicircle. Wait, no, the semicircle is the upper half of the circle. So, the chord PQ is horizontal at y = 0.75, and it's a chord of the circle. So, the two endpoints P and Q are located at ( -sqrt(r² - h²), h ) and ( sqrt(r² - h²), h ). Wait, but we already know that the length of PQ is 8 meters. The length of PQ is 2*sqrt(r² - h²). Wait, yes, that's the formula for chord length: 2*sqrt(r² - d²), where d is the distance from the center to the chord. So here, d = h = 0.75. So length PQ is 2*sqrt(r² - h²) = 8. Which gives sqrt(r² - h²) = 4, so r² - h² = 16. Which is consistent with our equation (1). Therefore, the endpoints of PQ are at ( -4, 0.75 ) and ( 4, 0.75 ). Similarly, chord MN is at distance h + 2 = 0.75 + 2 = 2.75 meters from the center (since the vertical distance between MN and PQ is 2 meters). The length of MN is 6 meters, so half-length is 3 meters. Therefore, the endpoints of MN are at ( -3, 2.75 ) and ( 3, 2.75 ).Wait, but hold on, the chord MN is at distance h + 2 = 2.75 from the center. So the y-coordinate of MN is 2.75. Then, the endpoints M and N are at ( -sqrt(r² - (2.75)^2 ), 2.75 ) and ( sqrt(r² - (2.75)^2 ), 2.75 ). Wait, but earlier we said that the length of MN is 6, so 2*sqrt(r² - (2.75)^2 ) = 6 ⇒ sqrt(r² - (2.75)^2 ) = 3 ⇒ r² - 7.5625 = 9 ⇒ r² = 16.5625. Which matches our previous calculation (since 0.75^2 + 16 = 0.5625 + 16 = 16.5625). Therefore, the endpoints of MN are at ( -3, 2.75 ) and ( 3, 2.75 ). Similarly, endpoints of PQ are at ( -4, 0.75 ) and ( 4, 0.75 ).Therefore, points M and P are ( -3, 2.75 ) and ( -4, 0.75 ), respectively. Wait, no. Wait, if MN is the upper chord with endpoints at (-3, 2.75) and (3, 2.75), then the left endpoint is M (-3, 2.75), right endpoint is N (3, 2.75). Similarly, PQ is the lower chord with endpoints at (-4, 0.75) and (4, 0.75), so left endpoint P (-4, 0.75), right endpoint Q (4, 0.75). So, arc MP connects point M (-3, 2.75) to point P (-4, 0.75) along the semicircle.Wait, but the semicircle is from (-r, 0) to (r, 0). Wait, but our center is at (0,0), and the semicircle is the upper half. So, points M (-3, 2.75), N (3, 2.75), P (-4, 0.75), Q (4, 0.75) are all on the semicircle. But wait, the coordinates must satisfy the equation x² + y² = r² = 16.5625.Let's check point M (-3, 2.75):x² + y² = 9 + (2.75)^2 = 9 + 7.5625 = 16.5625. Correct.Point P (-4, 0.75):x² + y² = 16 + 0.5625 = 16.5625. Correct.So, yes, these points are on the semicircle.Now, to find the arc length from M (-3, 2.75) to P (-4, 0.75). The arc length is r * θ, where θ is the central angle between points M and P in radians.To find θ, we can find the angle each point makes with the positive x-axis and subtract them.For point M (-3, 2.75):This is in the second quadrant (since x is negative, y is positive). The angle can be found using arctangent of y/x, but since it's in the second quadrant, we need to adjust accordingly.Let’s compute the angle α for point M:tan(α) = y / |x| = 2.75 / 3 ≈ 0.9167So, α = arctan(0.9167) ≈ 42.5 degrees. But since it's in the second quadrant, the angle from the positive x-axis is 180 - 42.5 = 137.5 degrees.Similarly, for point P (-4, 0.75):This is also in the second quadrant.tan(β) = y / |x| = 0.75 / 4 = 0.1875β = arctan(0.1875) ≈ 10.6 degrees. So, the angle from the positive x-axis is 180 - 10.6 = 169.4 degrees.Wait, but wait. Wait, angle from positive x-axis to point P is measured counterclockwise. For point P (-4, 0.75), which is in the second quadrant, the angle is π - arctan(0.75 / 4).Similarly, for point M (-3, 2.75), the angle is π - arctan(2.75 / 3).So, let's compute these angles in radians.First, compute angle for point M:x = -3, y = 2.75tan(θ_M) = y / |x| = 2.75 / 3 ≈ 0.916666...θ_M = π - arctan(0.916666...)Compute arctan(0.916666...):arctan(0.916666) ≈ 0.741 radians (since tan(0.741) ≈ 0.916). Let me check with calculator:tan(0.741) ≈ tan(42.5 degrees) ≈ 0.916, yes. 0.741 radians is approximately 42.5 degrees.So, θ_M = π - 0.741 ≈ 3.1416 - 0.741 ≈ 2.4006 radians.Similarly, for point P (-4, 0.75):tan(θ_P) = y / |x| = 0.75 / 4 = 0.1875arctan(0.1875) ≈ 0.185 radians (since tan(0.185) ≈ 0.187). Check:tan(10.6 degrees) ≈ 0.187, yes. 0.185 radians is approximately 10.6 degrees.Therefore, θ_P = π - 0.185 ≈ 3.1416 - 0.185 ≈ 2.9566 radians.Then, the central angle between points M and P is θ_P - θ_M ≈ 2.9566 - 2.4006 ≈ 0.556 radians.Therefore, the arc length MP is r * θ ≈ 4.0697 * 0.556 ≈ let's calculate:4.0697 * 0.5 = 2.034854.0697 * 0.056 ≈ 0.228So, total ≈ 2.03485 + 0.228 ≈ 2.26285 meters. Approximately 2.263 meters.Wait, but let me check these calculations more accurately.First, compute θ_M and θ_P more precisely.For point M (-3, 2.75):tan(α) = 2.75 / 3 ≈ 0.9166667α = arctan(0.9166667)Using a calculator, arctan(0.9166667) ≈ 0.741 radians (as before). So θ_M = π - 0.741 ≈ 2.4006 radians.For point P (-4, 0.75):tan(β) = 0.75 / 4 = 0.1875β = arctan(0.1875) ≈ 0.1854 radians. Therefore, θ_P = π - 0.1854 ≈ 2.9562 radians.Therefore, the angle between them is θ_P - θ_M ≈ 2.9562 - 2.4006 ≈ 0.5556 radians.Then, arc length is r * θ ≈ 4.0697 * 0.5556 ≈Compute 4 * 0.5556 = 2.22240.0697 * 0.5556 ≈ 0.0387Total ≈ 2.2224 + 0.0387 ≈ 2.2611 meters.So, approximately 2.261 meters. Rounded to 0.001 meters, that's 2.261 meters.But let's check this another way. Maybe using coordinates to find the angle.Another method: the central angle between two points can be found using the dot product formula.Given two points M and P on the circle, the angle θ between them is given by:cosθ = (M · P) / (r²)Where M · P is the dot product of the vectors OM and OP.Coordinates:Point M: (-3, 2.75)Point P: (-4, 0.75)Dot product M · P = (-3)(-4) + (2.75)(0.75) = 12 + 2.0625 = 14.0625r² = 16.5625Therefore, cosθ = 14.0625 / 16.5625 ≈ 0.849057Therefore, θ = arccos(0.849057) ≈ Let's compute that.arccos(0.849057). Let's recall that cos(0.556 radians) ≈ 0.849. Indeed, cos(0.556) ≈ cos(31.8 degrees) ≈ 0.849. So θ ≈ 0.556 radians, which matches our previous calculation.Therefore, the central angle is approximately 0.556 radians, so arc length is 4.0697 * 0.556 ≈ 2.261 meters.Therefore, the length of arc MP is approximately 2.261 meters. Rounded to three decimal places, 2.261 meters. But let me confirm once again.Wait, but let me check the exact calculation:r = sqrt(16.5625) ≈ 4.0697 metersθ ≈ 0.5556 radiansSo, 4.0697 * 0.5556:First compute 4 * 0.5556 = 2.22240.0697 * 0.5556 ≈ 0.0697 * 0.5 = 0.03485 and 0.0697 * 0.0556 ≈ 0.00387, total ≈ 0.03485 + 0.00387 ≈ 0.0387Total ≈ 2.2224 + 0.0387 ≈ 2.2611 meters. So, 2.261 meters.But let's check using more precise calculations:r = sqrt(16.5625). Wait, wait, 16.5625 is equal to 265/16. Wait, 16.5625 * 16 = 265. So, sqrt(265)/4 ≈ 16.2788/4 ≈ 4.0697. Correct.So, r = 4.0697 meters.θ = arccos(14.0625 / 16.5625) = arccos(0.8490566). Let's compute this using a calculator.Using a calculator: arccos(0.8490566). Let me convert this to degrees first. cos⁻¹(0.8490566) ≈ 31.8 degrees. Convert to radians: 31.8 * π / 180 ≈ 0.555 radians. So, θ ≈ 0.555 radians.Thus, arc length: 4.0697 * 0.555 ≈ 2.261 meters.But let me compute this more accurately.Compute θ:cosθ = 14.0625 / 16.5625 = 14.0625 ÷ 16.5625Let me compute this division:14.0625 ÷ 16.5625Multiply numerator and denominator by 16 to eliminate decimals:14.0625 * 16 = 22516.5625 * 16 = 265Therefore, cosθ = 225 / 265 ≈ 0.8490566Therefore, θ = arccos(225/265). Let's compute this using exact triangle.Let’s consider a right triangle where adjacent side is 225, hypotenuse is 265. Then, the opposite side would be sqrt(265² - 225²) = sqrt(70225 - 50625) = sqrt(19600) = 140.Therefore, cosθ = 225/265, sinθ = 140/265.Therefore, θ = arctan(140/225) = arctan(28/45) ≈ arctan(0.6222). Let's compute that.arctan(0.6222) ≈ 0.555 radians. Which matches previous results.Thus, θ ≈ 0.555 radians.Therefore, the arc length is r * θ = sqrt(265)/4 * arccos(225/265). However, since we have exact values here, maybe we can compute this more accurately.Wait, but we can just use the approximate values we have.Alternatively, use θ = arccos(225/265). Let's compute this using a calculator:θ = arccos(225/265) ≈ arccos(0.8490566) ≈ 0.55565 radians.Therefore, arc length ≈ 4.0697 * 0.55565 ≈4 * 0.55565 = 2.22260.0697 * 0.55565 ≈ 0.0387Total ≈ 2.2226 + 0.0387 ≈ 2.2613 meters.So, approximately 2.261 meters. Rounded to three decimal places, 2.261 meters.But let's check with another approach.Alternatively, since we know the coordinates of M and P, we can compute the angle between them using vector methods.Coordinates:M: (-3, 2.75)P: (-4, 0.75)Vectors OM and OP are (-3, 2.75) and (-4, 0.75). The angle between them can be found using the dot product formula:cosθ = (OM · OP) / (|OM| |OP|)We already computed OM · OP = 14.0625|OM| = |OP| = r = 4.0697Therefore, cosθ = 14.0625 / (4.0697^2) ≈ 14.0625 / 16.5625 ≈ 0.8490566, which is the same as before.Thus, θ ≈ 0.55565 radians, arc length ≈ 4.0697 * 0.55565 ≈ 2.2613 meters.Therefore, the answer is approximately 2.261 meters.But let me check if there is a mistake here. Because if we have points M (-3, 2.75) and P (-4, 0.75), the arc from M to P along the semicircle is the shorter arc connecting them, right? Since the semicircle is 180 degrees, the arc length should be less than π*r ≈ 12.8 meters. Our calculated 2.261 meters is reasonable.Alternatively, maybe the arc is the one going the other way around the semicircle? But since it's a semicircle, the only arc between M and P is the one that's 0.555 radians, which is about 31.8 degrees. That seems correct.Wait, but another way to check: the total circumference of the semicircle is π*r ≈ 3.1416*4.0697 ≈ 12.787 meters. If the arc MP is about 2.261 meters, then the remaining arc from P to M going the other way would be 12.787 - 2.261 ≈ 10.526 meters, which is much longer, so the 2.261 meters is indeed the shorter arc, which makes sense.Therefore, I think the length of arc MP is approximately 2.261 meters.But just to make sure, let's recap all steps:1. Found the radius r by using the chord lengths and their distances from the center.2. Determined coordinates of points M and P based on chord lengths and distances.3. Calculated the central angle between M and P using the dot product formula.4. Computed the arc length as r*θ.All steps check out, calculations seem consistent. Therefore, I think the answer is 2.261 meters.Final Answerboxed{2.261}</think>Given a semicircle with center ( O ), two parallel chords ( MN ) and ( PQ ) are present. The distance between these chords is 2 meters, ( MN = 6 ) meters, and ( PQ = 8 ) meters. We need to find the length of the arc ( overparen{MP} ).1. Determine the radius ( r ) of the semicircle: - Let ( h ) be the distance from the center ( O ) to the chord ( PQ ). - Using the chord length formula for ( PQ ): [ 8 = 2sqrt{r^2 - h^2} implies 4 = sqrt{r^2 - h^2} implies r^2 - h^2 = 16 quad (1) ] - For chord ( MN ), which is 2 meters above ( PQ ): [ 6 = 2sqrt{r^2 - (h + 2)^2} implies 3 = sqrt{r^2 - (h + 2)^2} implies 9 = r^2 - (h^2 + 4h + 4) implies r^2 - h^2 - 4h = 13 quad (2) ] - Solving equations (1) and (2): [ 16 - 4h = 13 implies 4h = 3 implies h = 0.75 text{ meters} ] [ r^2 = 0.75^2 + 16 = 0.5625 + 16 = 16.5625 implies r = sqrt{16.5625} approx 4.0697 text{ meters} ]2. Find the coordinates of points ( M ) and ( P ): - Chord ( MN ) is at ( y = 2.75 ) meters with endpoints ( (-3, 2.75) ) and ( (3, 2.75) ). - Chord ( PQ ) is at ( y = 0.75 ) meters with endpoints ( (-4, 0.75) ) and ( (4, 0.75) ). - Points ( M ) and ( P ) are ( (-3, 2.75) ) and ( (-4, 0.75) ) respectively.3. Calculate the central angle ( theta ) between points ( M ) and ( P ): - Using the dot product formula: [ cos theta = frac{(-3)(-4) + (2.75)(0.75)}{r^2} = frac{14.0625}{16.5625} approx 0.849057 ] [ theta = arccos(0.849057) approx 0.55565 text{ radians} ]4. Compute the arc length ( overparen{MP} ): - Arc length ( = r theta approx 4.0697 times 0.55565 approx 2.261 text{ meters} ).Thus, the length of the arc ( overparen{MP} ) is (boxed{2.261}) meters.]

🔑:- 1. As shown in Figure 6, draw a radius O T perpendicular to the parallel chords M N and P Q, with the feet of the perpendiculars being C and D, respectively. Connect O P and O M. Let the distance from the center O to P Q be s, and the radius of odot O be R. Then, in the right triangles triangle O P D and triangle O M C, we can obtain the system of equations:left{begin{array}{l}s^{2}+16=R^{2}, (s+2)^{2}+9=R^{2} .end{array}right.Solving these equations, we get s=frac{3}{4}, R=frac{sqrt{265}}{4}.Let angle T O M=alpha, angle T O P=beta. Thentan alpha=frac{3}{h+s}=frac{12}{11} Rightarrow alpha=arctan frac{12}{11},tan beta=frac{4}{s}=frac{16}{3} Rightarrow beta=arctan frac{16}{3}.Thus, the length of overparen{M P} is =R(beta-alpha)=frac{sqrt{265}}{4}left(arctan frac{16}{3}-arctan frac{12}{11}right)approx 2.26519511045.Therefore, the length of overparen{M P} is 2.265 mathrm{~m}.

🧠:[<think>Okay, so I have this problem here with two parts. Let me start with part (1). The function f(x) is given as sinx cosx + sqrt(3) cos²x - sqrt(3)/2. Then g(x) is f(x - θ), where θ is between 0 and π/2. The graph of g(x) is symmetric about the point (π/6, 0). I need to find θ.First, I need to recall that symmetry about a point means that the point is a center of symmetry. So, for any point (a, b) on the graph, the point (2π/6 - a, -b) should also be on the graph. Wait, the point is (π/6, 0), so if (x, y) is on the graph, then (2*(π/6) - x, -y) should be on the graph. That is, (π/3 - x, -y) is also a point on the graph. So, the function g satisfies g(π/3 - x) = -g(x). That's the condition for symmetry about (π/6, 0). Because the midpoint between x and π/3 - x is π/6, and the midpoint between y and -y is 0.So, the key equation here is g(π/3 - x) = -g(x). Since g(x) = f(x - θ), then substituting, we have f(π/3 - x - θ) = -f(x - θ). Let me replace x with x + θ to make it easier. Let me set y = x - θ. Then, x = y + θ. So, substituting into the equation:f(π/3 - (y + θ) - θ) = -f(y + θ - θ)Simplify the left side: π/3 - y - 2θ, and the right side is -f(y). So the equation becomes f(π/3 - y - 2θ) = -f(y).Therefore, for all y, f(π/3 - 2θ - y) = -f(y). So this is an identity that must hold for all y, which means that the function f is symmetric about the point ( (π/3 - 2θ)/2 , 0 ). Wait, but the function f is shifted by θ to get g, and g is symmetric about (π/6, 0). Hmm, maybe another approach is needed.Alternatively, since f(x - θ) = g(x), and g is symmetric about (π/6, 0), then for all x, g(π/3 - x) = -g(x). So replacing g with f shifted by θ:f(π/3 - x - θ) = -f(x - θ). Let's let u = x - θ. Then x = u + θ. Substitute into left side:f(π/3 - (u + θ) - θ) = f(π/3 - u - 2θ) = -f(u). So f(π/3 - 2θ - u) = -f(u) for all u. So f is symmetric about the point ( (π/3 - 2θ)/2, 0 ). But f itself must have a point of symmetry here. So to find θ, we need to determine where f has such a point of symmetry.Alternatively, perhaps f is an odd function shifted by some amount. Let me first simplify the given f(x):f(x) = sinx cosx + sqrt(3) cos²x - sqrt(3)/2.Maybe I can rewrite this using trigonometric identities. Let's see:sinx cosx is (1/2) sin2x. Then sqrt(3) cos²x can be written as sqrt(3) * (1 + cos2x)/2. So putting it all together:f(x) = (1/2) sin2x + sqrt(3)/2 (1 + cos2x) - sqrt(3)/2Simplify the constants: sqrt(3)/2 - sqrt(3)/2 cancels out. So we have:f(x) = (1/2) sin2x + (sqrt(3)/2) cos2xThis simplifies to:f(x) = sin(2x + φ), where φ is a phase shift. Because (1/2) sin2x + (sqrt(3)/2) cos2x is of the form A sin2x + B cos2x, which can be written as C sin(2x + φ).Compute the amplitude C: sqrt( (1/2)^2 + (sqrt(3)/2)^2 ) = sqrt(1/4 + 3/4) = sqrt(1) = 1. So C = 1. Then the phase φ is such that tanφ = B/A = (sqrt(3)/2)/(1/2) = sqrt(3). So φ = π/3. Therefore:f(x) = sin(2x + π/3)Wait, check:sin(2x + π/3) = sin2x cos(π/3) + cos2x sin(π/3) = (1/2) sin2x + (sqrt(3)/2) cos2x, which matches. Yes, so f(x) = sin(2x + π/3). Great, that simplifies things.So f(x) = sin(2x + π/3). Therefore, g(x) = f(x - θ) = sin(2(x - θ) + π/3) = sin(2x - 2θ + π/3).Now, the graph of g(x) is symmetric about the point (π/6, 0). For a function to be symmetric about a point (h, k), the general condition is that f(h + a) - k = -f(h - a) + k. Here, k is 0, so g(π/6 + a) = -g(π/6 - a) for all a.So, substituting into g(x):g(π/6 + a) = sin(2*(π/6 + a) - 2θ + π/3) = sin(π/3 + 2a - 2θ + π/3) = sin(2π/3 + 2a - 2θ)Similarly, -g(π/6 - a) = -sin(2*(π/6 - a) - 2θ + π/3) = -sin(π/3 - 2a - 2θ + π/3) = -sin(2π/3 - 2a - 2θ)For these to be equal for all a, we must have:sin(2π/3 + 2a - 2θ) = -sin(2π/3 - 2a - 2θ)Let me use the identity sin(A) = -sin(B) implies A = -B + 2πn or A = π + B + 2πn for some integer n. But since this must hold for all a, the coefficients of a must match. Let's write both sides:Left side: sin(2π/3 - 2θ + 2a)Right side: -sin(2π/3 - 2θ - 2a)Hmm, perhaps using angle addition formulas. Let me denote φ = 2π/3 - 2θ. Then:sin(φ + 2a) = -sin(φ - 2a)Expand both sides using sine addition:Left: sinφ cos2a + cosφ sin2aRight: -[sinφ cos2a - cosφ sin2a] = -sinφ cos2a + cosφ sin2aSet equal:sinφ cos2a + cosφ sin2a = -sinφ cos2a + cosφ sin2aSubtract cosφ sin2a from both sides:sinφ cos2a = -sinφ cos2aBring terms together:sinφ cos2a + sinφ cos2a = 0 => 2 sinφ cos2a = 0This must hold for all a. But cos2a is not identically zero, so sinφ must be zero. Therefore, sinφ = 0. But φ = 2π/3 - 2θ. Therefore:sin(2π/3 - 2θ) = 0Thus, 2π/3 - 2θ = nπ, n integerSolve for θ:2θ = 2π/3 - nπθ = π/3 - nπ/2But given that 0 < θ < π/2, let's find n such that θ is in that interval.For n=0: θ = π/3 ≈ 1.047, which is less than π/2 ≈ 1.570. So θ = π/3 is valid.For n=1: θ = π/3 - π/2 = -π/6, which is negative, invalid.For n=-1: θ = π/3 + π/2 = 5π/6, which is greater than π/2, invalid.Thus, the only valid solution is θ = π/3. But let's verify this.If θ = π/3, then g(x) = sin(2x - 2*(π/3) + π/3) = sin(2x - π/3). So g(x) = sin(2x - π/3). Let's check symmetry about (π/6, 0).Compute g(π/6 + a) = sin(2*(π/6 + a) - π/3) = sin(π/3 + 2a - π/3) = sin(2a)Compute -g(π/6 - a) = -sin(2*(π/6 - a) - π/3) = -sin(π/3 - 2a - π/3) = -sin(-2a) = sin(2a)Thus, g(π/6 + a) = sin(2a) and -g(π/6 - a) = sin(2a), so they are equal. Therefore, θ = π/3 is correct.Wait, but hold on. If θ = π/3, then indeed the symmetry condition is satisfied. So θ is π/3. But the answer is supposed to be in 0 < θ < π/2, which π/3 is approximately 1.047, which is less than π/2 (≈1.570). So that's okay.But wait, let me check again. Let me compute θ:From 2π/3 - 2θ = nπ.We had 2θ = 2π/3 - nπ => θ = π/3 - nπ/2.For n=0, θ=π/3.For n=1, θ=π/3 - π/2= -π/6 invalid.For n=-1, θ=π/3 + π/2=5π/6 invalid.Therefore, only θ=π/3 is valid. So answer for part (1) is π/3.Wait, but let me make sure. Maybe the original approach with the function f being symmetric about some point? Wait, f(x) is sin(2x + π/3). Let's see if f(x) has any symmetry. For example, is f symmetric about some point?Suppose f(h - x) = -f(h + x). Then the function is symmetric about (h, 0). Let's see:sin(2(h - x) + π/3) = -sin(2(h + x) + π/3)Let me expand both sides:Left: sin(2h - 2x + π/3)Right: -sin(2h + 2x + π/3)So we need sin(2h - 2x + π/3) = -sin(2h + 2x + π/3)Use identity sin(A) = -sin(B) => A = -B + 2πn or A = π + B + 2πnBut this must hold for all x. Let's set coefficients of x equal.Looking at arguments:Left: (-2x) + (2h + π/3)Right: -(2x) - (2h + π/3) + π ?Wait, maybe another approach. Let me use the identity sin(A) + sin(B) = 2 sin( (A+B)/2 ) cos( (A-B)/2 )But here we have sin(2h - 2x + π/3) + sin(2h + 2x + π/3) = 0Using the identity:2 sin( (2h - 2x + π/3 + 2h + 2x + π/3)/2 ) cos( (2h - 2x + π/3 - (2h + 2x + π/3))/2 ) = 0Simplify:2 sin( (4h + 2π/3)/2 ) cos( (-4x)/2 ) = 0Which is:2 sin(2h + π/3) cos(-2x) = 0Since cos(-2x)=cos(2x), this must hold for all x. The only way this can be true for all x is if sin(2h + π/3) = 0.Therefore, sin(2h + π/3) = 0 => 2h + π/3 = nπ => h = (nπ - π/3)/2Thus, the function f(x) is symmetric about points (h,0) where h = (nπ - π/3)/2. For example, n=1: h=(π - π/3)/2=(2π/3)/2=π/3. So f(x) is symmetric about (π/3, 0). Similarly, n=0: h=(-π/3)/2=-π/6, which is not in our interval.Therefore, the original function f(x) is symmetric about (π/3, 0). Therefore, if we shift f(x) to get g(x), such that g(x) is symmetric about (π/6,0), then we need to shift f(x) left or right so that its center of symmetry moves from π/3 to π/6.Since g(x) = f(x - θ), shifting f(x) to the right by θ. The original center of f(x) is at x=π/3. After shifting right by θ, the new center is at x=π/3 + θ. But we want the new center to be at π/6. Therefore:π/3 + θ = π/6 => θ = π/6 - π/3 = -π/6. But θ is supposed to be between 0 and π/2, so this is invalid. Wait, that's a problem.Alternatively, maybe shifting f(x) left by θ, but the problem says g(x) = f(x - θ), which is shifting right by θ. Wait, so if the original center is at π/3, shifting right by θ moves the center to π/3 + θ. We need it to be at π/6. So:π/3 + θ = π/6 => θ = π/6 - π/3 = -π/6. Not possible.But that contradicts our earlier result where θ=π/3. Hmmm. So there must be something wrong here.Wait, perhaps my assumption that shifting f(x) right by θ moves the center to the right by θ is incorrect? Wait, no. If you have a function f(x) with a center at (h, k), then f(x - θ) would shift the graph right by θ, so the new center is at (h + θ, k). Therefore, original center of f(x) is (π/3, 0). Then shifting right by θ would move the center to (π/3 + θ, 0). We need this center to be (π/6, 0). Therefore:π/3 + θ = π/6 => θ = π/6 - π/3 = -π/6. Not possible. But we found θ=π/3 earlier, which conflicts.Wait, so there's a contradiction here. Which one is correct?Earlier, we derived θ=π/3 by using the symmetry condition, and verified that with θ=π/3, the function g(x)=sin(2x - π/3) is symmetric about (π/6,0). Let's check that again.Compute g(π/6 + a) = sin(2*(π/6 + a) - π/3) = sin(π/3 + 2a - π/3) = sin(2a)Compute -g(π/6 - a) = -sin(2*(π/6 - a) - π/3) = -sin(π/3 - 2a - π/3) = -sin(-2a) = sin(2a)Thus, they are equal. Therefore, θ=π/3 is indeed correct.But according to the shifting idea, the center should be at π/3 + θ, which would be π/3 + π/3 = 2π/3. But in reality, the center is at π/6. So how does that reconcile?Wait, perhaps the mistake is in the assumption that the original center is π/3. Wait, when we derived that f(x) is symmetric about (π/3,0), but actually, in the process above, we saw that for f(x) = sin(2x + π/3), which is a sine function with phase shift.Wait, the general sine function sin(Bx + C) has its center at -C/B, but for symmetry points, the function sin(Bx + C) is symmetric about its midline, which for basic sine function is at y=0. However, the points of symmetry for a sinusoidal function are at its points of inflection. Wait, maybe this is more complicated.Wait, a standard sine function, sin(x), is symmetric about every point of the form (kπ, 0). So, for example, (0,0), (π,0), etc. Because sin(π - x) = sin(x), but sin(π + x) = -sin(x). Wait, no. Wait, sin(π - x) = sinx, which is a reflection over x=π/2. Hmm, perhaps the function sin(2x + π/3) has symmetry about certain points.Alternatively, perhaps when we derived the condition for f, we considered the equation f(π/3 - 2θ - u) = -f(u). So, if f is symmetric about some point (h, 0), then f(2h - u) = -f(u). Comparing to the equation f(π/3 - 2θ - u) = -f(u), then 2h - u = π/3 - 2θ - u => 2h = π/3 - 2θ => h = (π/3 - 2θ)/2. But if f is symmetric about its own center (π/3,0), then h = π/3. Therefore:(π/3 - 2θ)/2 = π/3 => π/3 - 2θ = 2π/3 => -2θ = π/3 => θ = -π/6. Which is invalid. So this approach seems contradictory.But since we already verified θ=π/3 works, perhaps the problem is that f(x) is not symmetric about (π/3,0) even though we thought it was. Wait, let's test f(π/3 - x) vs -f(x). Let's compute f(π/3 - x):f(π/3 - x) = sin(2(π/3 - x) + π/3) = sin(2π/3 - 2x + π/3) = sin(π - 2x) = sin2x. Because sin(π - 2x) = sin2x. Wait, but -f(x) = -sin(2x + π/3). So f(π/3 - x) = sin2x, which is not equal to -sin(2x + π/3). So they are not negatives. Therefore, f is not symmetric about (π/3, 0). Wait, that's confusing. Then why did earlier steps suggest that?Wait, maybe my mistake was in the step where I thought f(x) is symmetric about (π/3, 0). When we set the equation sin(2h + π/3) = 0, giving h=(nπ - π/3)/2, but perhaps the actual symmetry is not about (h,0) but some other point. Let me check.Wait, suppose we have a function f(x) = sin(2x + π/3). Let's find a point (h, 0) such that f(h + a) = -f(h - a) for all a. Then:sin(2(h + a) + π/3) = -sin(2(h - a) + π/3)Let me expand both sides:Left: sin(2h + 2a + π/3)Right: -sin(2h - 2a + π/3)So, sin(2h + 2a + π/3) + sin(2h - 2a + π/3) = 0Using the identity sin A + sin B = 2 sin( (A+B)/2 ) cos( (A-B)/2 )So:2 sin( (2h + 2a + π/3 + 2h - 2a + π/3)/2 ) cos( (2h + 2a + π/3 - (2h - 2a + π/3))/2 ) = 0Simplify:2 sin( (4h + 2π/3)/2 ) cos( (4a)/2 ) = 0Which is:2 sin(2h + π/3) cos(2a) = 0For this to hold for all a, cos(2a) term can't be always zero, so sin(2h + π/3) must be zero. Therefore:sin(2h + π/3) = 0 => 2h + π/3 = nπ => h = (nπ - π/3)/2So possible h's are h=(nπ - π/3)/2. For example, n=1: h=(π - π/3)/2=(2π/3)/2=π/3. So h=π/3. Let's check if f(π/3 + a) = -f(π/3 - a):f(π/3 + a) = sin(2*(π/3 + a) + π/3) = sin(2π/3 + 2a + π/3) = sin(π + 2a) = -sin2af(π/3 - a) = sin(2*(π/3 - a) + π/3) = sin(2π/3 - 2a + π/3) = sin(π - 2a) = sin2aSo indeed, f(π/3 + a) = -sin2a and -f(π/3 - a) = -sin2a. Therefore, f(π/3 + a) = -f(π/3 - a). Therefore, f(x) is symmetric about (π/3,0). Therefore, my initial assertion was correct. So f(x) is symmetric about (π/3,0).But then g(x) = f(x - θ) should be symmetric about (π/3 + θ, 0). But the problem states that g(x) is symmetric about (π/6, 0). Therefore, setting π/3 + θ = π/6 => θ = π/6 - π/3 = -π/6. But θ must be between 0 and π/2. So this is impossible. Contradiction.But earlier steps using the symmetry condition directly on g(x) gave θ=π/3, which works. So there must be a mistake here. Wait, perhaps shifting f(x) right by θ actually shifts the center to the left by θ? Wait, no, shifting a function right by θ moves all points to the right by θ. So if the original center is at π/3, then shifting right by θ would move the center to π/3 + θ. But we need the center at π/6, so π/3 + θ = π/6 => θ = -π/6. Not possible. So how come when we computed θ=π/3 it worked?Wait, perhaps the error is that in the original symmetry condition, when we set g(π/3 - x) = -g(x), but the center of symmetry for g(x) is (π/6, 0), so the correct condition is g(π/3 - x) = -g(x). Wait, the center is (h, k) = (π/6, 0). So for any x, g(2h - x) - k = -g(x) + k. Since k=0, this is g(2*(π/6) - x) = -g(x), which is g(π/3 - x) = -g(x). So that is correct.But since g(x) = f(x - θ), then substituting:f(π/3 - x - θ) = -f(x - θ)Let me let u = x - θ, so x = u + θ. Then:f(π/3 - (u + θ) - θ) = f(π/3 - u - 2θ) = -f(u)So, we have f(π/3 - 2θ - u) = -f(u). But earlier we determined that f is symmetric about (π/3,0), i.e., f(π/3 - u) = -f(π/3 + u). Wait, but here we have f(π/3 - 2θ - u) = -f(u). If we set π/3 - 2θ - u = π/3 + u', then maybe?Alternatively, since f(π/3 - u) = -f(π/3 + u), then if we set π/3 - 2θ - u = π/3 + u, then:π/3 - 2θ - u = π/3 + u => -2θ - u = u => -2θ = 2u => θ = -u.But u is a variable, so this can't hold for all u. Therefore, this approach is invalid.Alternatively, since f(π/3 - 2θ - u) = -f(u) must hold for all u, we can compare this to the known symmetry of f about π/3. Since f(π/3 - u) = -f(π/3 + u), let's set π/3 - 2θ - u = π/3 + v, then:π/3 - 2θ - u = π/3 + v => -2θ - u = vSo f(π/3 + v) = -f(u). But from the symmetry of f, f(π/3 + v) = -f(π/3 - v). Therefore:-f(π/3 - v) = -f(u) => f(π/3 - v) = f(u)But since v = -2θ - u, substituting:f(π/3 - (-2θ - u)) = f(π/3 + 2θ + u) = f(u)So we need f(π/3 + 2θ + u) = f(u) for all u. Therefore, the function f must be periodic with period π/3 + 2θ. But f(x) = sin(2x + π/3) has period π. Therefore:π/3 + 2θ = π => 2θ = 2π/3 => θ=π/3. Which matches our previous result.Therefore, this condition is satisfied when θ=π/3. So this is the answer. So even though shifting f(x) by θ=π/3 to the right would naively shift the center to π/3 + π/3=2π/3, but because of the periodicity and the symmetry condition, it works out.So, the answer to part (1) is θ=π/3.Now, moving to part (2). We have a triangle ABC with sides a, b, c opposite angles A, B, C respectively. The circumradius is R = sqrt(21)/3. Given that g(A/8) = -1/2, where g is the function from part (1), which we found θ=π/3, so g(x)=sin(2x - π/3). Also, side b=1, and we need to find the length of AD, where D is the trisection point of BC closer to B.First, let's recall that in any triangle, the Law of Sines holds: a/sinA = b/sinB = c/sinC = 2R. Given that R = sqrt(21)/3 and b=1, we can relate angle B to side b.Given g(A/8) = -1/2, and g(x)=sin(2x - π/3). Therefore:sin(2*(A/8) - π/3) = -1/2Simplify:sin(A/4 - π/3) = -1/2We need to solve for angle A. Let's find all possible solutions:sinθ = -1/2 has solutions θ = 7π/6 + 2πn or θ = 11π/6 + 2πn, where n is integer.But since A is an angle in a triangle, 0 < A < π, so A/4 - π/3 must be within -π/3 < A/4 - π/3 < π/4 - π/3 = -π/12. So θ = A/4 - π/3 is in (-π/3, -π/12). So looking for θ in this interval where sinθ = -1/2.The reference angle for sinθ = 1/2 is π/6. So sinθ = -1/2 occurs at θ = 7π/6 and 11π/6, but considering θ must be in (-π/3, -π/12), which is between approximately -1.047 and -0.2618 radians. Let's convert 7π/6 ≈3.665 radians, which is outside the interval. 11π/6≈5.759 radians, also outside. But perhaps considering negative angles:θ = -π/6 (which is -0.523 radians) is in (-π/3 ≈ -1.047, -π/12 ≈ -0.2618)? No, -π/6 ≈ -0.523 is between -1.047 and -0.2618? Wait, -π/6 is approximately -0.523, which is within -π/3 ≈ -1.047 and -π/12 ≈ -0.2618. Wait, -π/12 is approximately -0.2618, which is greater (less negative) than -π/6. So θ = -π/6 is within the interval.But wait, θ = A/4 - π/3 = -π/6 => A/4 = π/3 - π/6 = π/6 => A = (π/6)*4 = 2π/3. So A = 2π/3 ≈ 120 degrees. Let's check if this is the only solution.Another solution for sinθ = -1/2 in the interval (-π/3, -π/12) could be θ = -5π/6, but that's -2.618 radians, which is less than -π/3 (-1.047), so outside the interval. So the only solution is θ = -π/6. Therefore, A/4 - π/3 = -π/6 => A/4 = π/3 - π/6 = π/6 => A = 4*(π/6) = 2π/3. So angle A is 120 degrees or 2π/3 radians.So angle A = 2π/3. Now, we can find the other angles and sides using the Law of Sines.Given b=1, which is the side opposite angle B. The Law of Sines states that a/sinA = b/sinB = c/sinC = 2R. Given R = sqrt(21)/3, so 2R = 2*sqrt(21)/3.Therefore:b/sinB = 2*sqrt(21)/3 => 1/sinB = 2*sqrt(21)/3 => sinB = 3/(2*sqrt(21)) = (3)/(2*sqrt(21)) = rationalize: (3*sqrt(21))/(2*21) = sqrt(21)/14.Wait, 3/(2*sqrt(21)) = multiply numerator and denominator by sqrt(21):3*sqrt(21)/(2*21) = sqrt(21)/14. So sinB = sqrt(21)/14.Therefore, angle B is arcsin(sqrt(21)/14). Let me compute this value.But maybe we can find angle B in terms of other angles. Since the triangle has angles A + B + C = π. A = 2π/3, so B + C = π - 2π/3 = π/3. Let's denote angle B = β, angle C = π/3 - β.Now, using the Law of Sines:a / sinA = 2R => a = 2R sinA = 2*(sqrt(21)/3)*sin(2π/3) = 2*(sqrt(21)/3)*(sqrt(3)/2) = (sqrt(21)*sqrt(3))/3 = sqrt(63)/3 = 3*sqrt(7)/3 = sqrt(7). So a = sqrt(7).Similarly, b=1, which is given.For side c:c = 2R sinC = 2*(sqrt(21)/3)*sin(π/3 - β). But maybe we can find sinC.Alternatively, since angle C = π/3 - β, and we can find sinC = sin(π/3 - β) = sinπ/3 cosβ - cosπ/3 sinβ = (sqrt(3)/2)cosβ - (1/2)sinβ.But we need to find β first. From sinB = sqrt(21)/14, and angle B = β.So β = arcsin(sqrt(21)/14). Let's compute cosβ.cosβ = sqrt(1 - (sqrt(21)/14)^2) = sqrt(1 - 21/196) = sqrt(175/196) = sqrt(175)/14 = 5*sqrt(7)/14.Therefore, sinC = (sqrt(3)/2)*(5sqrt(7)/14) - (1/2)*(sqrt(21)/14)Simplify:= (5 sqrt(21))/28 - sqrt(21)/28 = (5 sqrt(21) - sqrt(21))/28 = (4 sqrt(21))/28 = sqrt(21)/7Therefore, c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/(3*7)) = 2*(3/7) wait, sqrt(21)*sqrt(21) = 21.Therefore, 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/(3*7)) = 2*(21/21) = 2*1 = 2. Therefore, c=2.So sides are a = sqrt(7), b=1, c=2.Wait, let me confirm:Given sinC = sqrt(21)/7, then c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/(3*7)) = 2*(1) = 2. Correct.So sides:a = sqrt(7), opposite angle A=2π/3b=1, opposite angle B=arcsin(sqrt(21)/14)c=2, opposite angle C=π/3 - BNow, we need to find the length of AD, where D is the trisection point of BC closer to B. So BC is side a, which is sqrt(7). Wait, no, sides opposite angles: side a is opposite angle A, so BC is side a. Wait, in triangle ABC, side a is BC, side b is AC, and side c is AB.Wait, standard notation: side a is BC, opposite angle A; side b is AC, opposite angle B; side c is AB, opposite angle C.Therefore, BC is side a = sqrt(7). Point D is the trisection point of BC closer to B, so BD = (1/3) BC = sqrt(7)/3, DC = (2/3) BC = 2sqrt(7)/3.We need to find the length of AD.To find AD, we can use coordinates or vectors, but perhaps using the Law of Cosines in triangle ABD or ADC.Alternatively, use coordinates. Let me set up coordinate system.Let me place point B at the origin (0,0), point C at (sqrt(7), 0). Then point D is located at BD = sqrt(7)/3 along BC, so coordinates of D are (sqrt(7)/3, 0).Point A is somewhere in the plane. We need to find coordinates of A. To do that, we can use the fact that side AB = c=2, and AC = b=1.Given points B(0,0), C(sqrt(7), 0). Let point A be (x,y). Then:Distance from A to B: sqrt((x-0)^2 + (y-0)^2) = 2 => x² + y² = 4.Distance from A to C: sqrt((x - sqrt(7))² + y²) =1 => (x - sqrt(7))² + y² =1.Subtract the first equation from the second:(x - sqrt(7))² + y² - (x² + y²) =1 -4 => x² - 2sqrt(7)x +7 + y² -x² - y² = -3 => -2sqrt(7)x +7 = -3 => -2sqrt(7)x = -10 => x = (-10)/(-2sqrt(7)) =5/sqrt(7) =5sqrt(7)/7.Then substitute x into x² + y² =4:(25*7)/49 + y² =4 => 25/7 + y²=4 => y²=4 -25/7=28/7 -25/7=3/7 => y=±sqrt(3/7)=±sqrt(21)/7.But since the triangle has angles with A=2π/3, which is obtuse, so point A should be above the x-axis. So take y=sqrt(21)/7.Therefore, coordinates of A are (5sqrt(7)/7, sqrt(21)/7).Now, coordinates of D are (sqrt(7)/3, 0). So vector AD is from A(5sqrt(7)/7, sqrt(21)/7) to D(sqrt(7)/3, 0). Let's compute the distance between these two points.First, compute the differences in coordinates:Δx = sqrt(7)/3 -5sqrt(7)/7 = sqrt(7)(1/3 -5/7) = sqrt(7)( (7 -15)/21 ) = sqrt(7)(-8/21) = -8sqrt(7)/21Δy = 0 - sqrt(21)/7 = -sqrt(21)/7Then, the distance AD is sqrt( (Δx)^2 + (Δy)^2 )Compute (Δx)^2:(-8sqrt(7)/21)^2 = 64*7 / (441) = 448 / 441 = 64*7 / 63*7 = 64/63Wait, 8^2=64, 7*(8^2)=64*7=448. 21^2=441. So 448/441 = 64/63. Because 448 ÷7=64, 441 ÷7=63.(Δy)^2 = ( -sqrt(21)/7 )^2 =21/49=3/7.Wait, 21/49=3/7. Yes.Therefore, AD^2 = 64/63 +3/7 =64/63 +27/63=91/63=13/9. Therefore, AD= sqrt(13/9)=sqrt(13)/3.Therefore, the length of AD is sqrt(13)/3.But let me verify the calculation again to be sure.Δx = sqrt(7)/3 -5sqrt(7)/7. Let's compute this:sqrt(7)/3 -5sqrt(7)/7 = sqrt(7)(1/3 -5/7) = sqrt(7)( (7 -15)/21 )= sqrt(7)(-8/21)= -8sqrt(7)/21.Δx squared: ( -8sqrt(7)/21 )² =64*7 / (441)= 448 / 441=64/63.Δy squared: ( -sqrt(21)/7 )²=21/49=3/7.Convert 3/7 to 27/63? Wait, 3/7 =27/63? No, 3/7=27/63 is incorrect. 3/7= (3*9)/(7*9)=27/63. Wait, but 3*9=27 and 7*9=63. Yes, so 3/7=27/63.Therefore, AD²=64/63 +27/63=91/63=13/9. So AD= sqrt(13)/3. Yes, correct.Therefore, the answer is sqrt(13)/3.Final Answer(1) boxed{dfrac{pi}{3}} (2) boxed{dfrac{sqrt{13}}{3}}</think>Given the function ( f(x) = sin x cos x + sqrt{3} cos^2 x - frac{sqrt{3}}{2} ), we simplify it using trigonometric identities:[f(x) = frac{1}{2} sin 2x + frac{sqrt{3}}{2} cos 2x = sin left( 2x + frac{pi}{3} right)]The function ( g(x) = f(x - theta) ) is given, and we need to find ( theta ) such that the graph of ( g(x) ) is symmetric about the point ( left( frac{pi}{6}, 0 right) ). This implies ( gleft( frac{pi}{3} - x right) = -g(x) ). Substituting ( g(x) = sin left( 2(x - theta) + frac{pi}{3} right) ), we get:[sin left( 2 left( frac{pi}{3} - x right) - 2theta + frac{pi}{3} right) = -sin left( 2(x - theta) + frac{pi}{3} right)]Simplifying, we find that ( theta = frac{pi}{3} ).For part (2), given ( gleft( frac{A}{8} right) = -frac{1}{2} ), we solve:[sin left( 2 left( frac{A}{8} right) - frac{pi}{3} right) = -frac{1}{2}]This gives ( A = frac{2pi}{3} ). Using the Law of Sines with ( R = frac{sqrt{21}}{3} ) and ( b = 1 ), we find the sides ( a = sqrt{7} ), ( b = 1 ), and ( c = 2 ). Point ( D ) is the trisection point of ( BC ) closer to ( B ), so ( BD = frac{sqrt{7}}{3} ). Using coordinates, we find the length ( AD ) as:[AD = sqrt{left( frac{5sqrt{7}}{7} - frac{sqrt{7}}{3} right)^2 + left( frac{sqrt{21}}{7} - 0 right)^2} = frac{sqrt{13}}{3}] Final Answer(1) boxed{dfrac{pi}{3}} (2) boxed{dfrac{sqrt{13}}{3}},<think>Alright, so I have this problem here with two parts. Let me start with part (1). We have the function f(x) = sinx cosx + sqrt(3) cos²x - sqrt(3)/2. Then g(x) is defined as f(x - θ), where θ is between 0 and π/2. The graph of g(x) is symmetric about the point (π/6, 0). I need to find θ.First, let me recall what it means for a graph to be symmetric about a point. If a graph is symmetric about the point (a, b), then for any point (x, y) on the graph, the point (2a - x, 2b - y) is also on the graph. In this case, symmetry about (π/6, 0) means that for any x, g(2*(π/6) - x) = -g(x). So, g(π/3 - x) = -g(x). That's the condition we need to satisfy.Since g(x) = f(x - θ), then substituting into the symmetry condition:g(π/3 - x) = f(π/3 - x - θ) = -g(x) = -f(x - θ)So, f(π/3 - x - θ) = -f(x - θ). Let me denote y = x - θ. Then, the left side becomes f(π/3 - (y + θ) - θ) = f(π/3 - y - 2θ). So, f(π/3 - y - 2θ) = -f(y). Therefore, the function f satisfies f(π/3 - y - 2θ) = -f(y) for all y. This suggests that f has a point of symmetry at some point, but translated. Maybe f has a certain periodicity or symmetry that we can exploit.Alternatively, perhaps we can express f(x) in a simpler form using trigonometric identities, which might make it easier to analyze the symmetry.Let's try to simplify f(x):f(x) = sinx cosx + sqrt(3) cos²x - sqrt(3)/2.First, sinx cosx can be written as (1/2) sin2x. Then, cos²x can be written as (1 + cos2x)/2. Let me substitute these:f(x) = (1/2) sin2x + sqrt(3)*( (1 + cos2x)/2 ) - sqrt(3)/2.Simplify term by term:First term: (1/2) sin2xSecond term: sqrt(3)/2 (1 + cos2x) = sqrt(3)/2 + sqrt(3)/2 cos2xThird term: - sqrt(3)/2Combine the second and third terms: sqrt(3)/2 + sqrt(3)/2 cos2x - sqrt(3)/2 = sqrt(3)/2 cos2xTherefore, f(x) simplifies to (1/2) sin2x + (sqrt(3)/2) cos2xSo f(x) = (1/2) sin2x + (sqrt(3)/2) cos2xHmm, that looks like a single sinusoidal function. Remember that A sinθ + B cosθ can be written as C sin(θ + φ) or C cos(θ + φ). Let's see:(1/2) sin2x + (sqrt(3)/2) cos2xLet me factor out 1/2:(1/2)[ sin2x + sqrt(3) cos2x ]Let me recall that sin(a + b) = sin a cos b + cos a sin b. So, if we can express sin2x + sqrt(3) cos2x as a sine function with some phase shift.The general identity is A sinθ + B cosθ = C sin(θ + φ), where C = sqrt(A² + B²), and tanφ = B/A. Wait, actually, more precisely:A sinθ + B cosθ = C sin(θ + φ), where C = sqrt(A² + B²), and φ = arctan(B/A) if A ≠ 0.Alternatively, it can be written as C cos(θ - φ'), where φ' is another angle.Let me compute the amplitude first:C = sqrt( (1)^2 + (sqrt(3))^2 ) = sqrt(1 + 3) = sqrt(4) = 2. Wait, but in our case, the coefficients are 1 and sqrt(3) multiplied by 1/2. Wait, no, the expression is (1/2)[ sin2x + sqrt(3) cos2x ], so inside the brackets, A = 1 and B = sqrt(3). So the amplitude is 2, multiplied by 1/2 gives 1. So the amplitude is 1. Let's confirm:Inside the brackets: sin2x + sqrt(3) cos2x. So C = sqrt(1 + 3) = 2, so this expression is 2 sin(2x + φ). Then multiplied by 1/2 gives sin(2x + φ). Let's compute φ.tanφ = B/A = sqrt(3)/1 = sqrt(3). So φ = π/3. Therefore:sin2x + sqrt(3) cos2x = 2 sin(2x + π/3). Therefore, multiplying by 1/2 gives sin(2x + π/3). Wait, hold on:Wait, actually, the formula is A sinθ + B cosθ = C sin(θ + φ), where C = sqrt(A² + B²) and φ is such that cosφ = A/C and sinφ = B/C. Wait, actually, maybe I need to adjust. Let me check:Let me recall that:A sinθ + B cosθ = C sin(θ + φ), where:C = sqrt(A² + B²)sinφ = B/Ccosφ = A/CSo here, A = 1, B = sqrt(3). Therefore,C = sqrt(1 + 3) = 2sinφ = sqrt(3)/2, cosφ = 1/2. Therefore, φ = π/3.Hence,sin2x + sqrt(3) cos2x = 2 sin(2x + π/3). Therefore, multiplying by 1/2:f(x) = (1/2)(2 sin(2x + π/3)) = sin(2x + π/3). Wait, that seems too straightforward. Wait, but let me verify:If we expand sin(2x + π/3), we get sin2x cos(π/3) + cos2x sin(π/3) = sin2x*(1/2) + cos2x*(sqrt(3)/2), which is exactly (1/2) sin2x + (sqrt(3)/2) cos2x. So, yes, f(x) = sin(2x + π/3). Therefore, f(x) = sin(2x + π/3). So that's a simplified form.Therefore, f(x) = sin(2x + π/3). Then, g(x) = f(x - θ) = sin(2(x - θ) + π/3) = sin(2x - 2θ + π/3).So g(x) = sin(2x - 2θ + π/3). Now, the graph of g(x) is symmetric about the point (π/6, 0). So, as per the symmetry condition, for any x, g(π/3 - x) = -g(x).Wait, but let's recall that the symmetry about a point (a, b) means that for any point (x, y) on the graph, the point (2a - x, 2b - y) is also on the graph. Here, a = π/6, b = 0. Therefore, the condition is that for all x, g(2*(π/6) - x) = -g(x) => g(π/3 - x) = -g(x). So this must hold for all x.So, substituting into g(x):g(π/3 - x) = sin[2*(π/3 - x) - 2θ + π/3] = sin[(2π/3 - 2x) - 2θ + π/3] = sin(π - 2x - 2θ)On the other hand, -g(x) = -sin(2x - 2θ + π/3)So, we have sin(π - 2x - 2θ) = -sin(2x - 2θ + π/3)Using the identity sin(π - α) = sinα, so sin(π - 2x - 2θ) = sin(2x + 2θ). Wait, hold on. Let me compute that again.Wait, π - 2x - 2θ is the argument. So sin(π - A) = sinA if A is replaced by π - A. Wait, sin(π - A) = sinA. So sin(π - (2x + 2θ)) = sin(2x + 2θ). Wait, but here the argument is π - 2x - 2θ = π - (2x + 2θ). Therefore, sin(π - (2x + 2θ)) = sin(2x + 2θ). So, the left-hand side is sin(2x + 2θ), and the right-hand side is -sin(2x - 2θ + π/3). Therefore:sin(2x + 2θ) = -sin(2x - 2θ + π/3)This equation must hold for all x. Therefore, their arguments must satisfy certain relationships. Let's recall that sinA = -sinB implies that A = -B + 2πn or A = π + B + 2πn for some integer n. Since this must hold for all x, the coefficients of x must match, and the constants must satisfy the equation.Let's denote the left-hand side argument as A = 2x + 2θThe right-hand side argument inside sine is B = 2x - 2θ + π/3, and we have sinA = -sinB.So, sinA = -sinB => sinA + sinB = 0 => 2 sin( (A + B)/2 ) cos( (A - B)/2 ) = 0But since this has to hold for all x, the only way this can happen is if either sin( (A + B)/2 ) = 0 for all x, which is impossible unless the coefficient of x is zero, or cos( (A - B)/2 ) = 0 for all x, which similarly requires the coefficient of x in (A - B)/2 to be zero.But let's instead use the identity sinA = -sinB. So, sinA = -sinB implies that:Case 1: A = -B + 2πnCase 2: A = π + B + 2πnfor some integer n. Let's analyze both cases.Case 1: 2x + 2θ = - (2x - 2θ + π/3) + 2πnSimplify:2x + 2θ = -2x + 2θ - π/3 + 2πnBring terms with x to left and constants to right:2x + 2x = 2θ - π/3 + 2πn - 2θ4x = -π/3 + 2πnBut this must hold for all x, which is only possible if 4 = 0, which is impossible. So Case 1 is invalid.Case 2: 2x + 2θ = π + 2x - 2θ + π/3 + 2πnSimplify:Left side: 2x + 2θRight side: π + 2x - 2θ + π/3 + 2πnSubtract 2x from both sides:2θ = π - 2θ + π/3 + 2πnBring terms with θ to left and constants to right:2θ + 2θ = π + π/3 + 2πn4θ = (4π)/3 + 2πnTherefore:θ = ( (4π)/3 + 2πn ) / 4 = (π/3)(1 + (3/2)n )But θ is given to be between 0 and π/2. Let's see possible integer values of n.n is an integer. Let's consider n=0:θ = (π/3)(1 + 0 ) = π/3 ≈ 1.047, but π/2 ≈ 1.570, so π/3 is less than π/2. So θ = π/3 is possible. But we need to check if this is the only solution.Wait, but n can also be negative. Let's check n=-1:θ = (4π/3 + 2π*(-1)) /4 = (4π/3 - 2π)/4 = (4π/3 - 6π/3)/4 = (-2π/3)/4 = -π/6. Negative, which is not in the interval (0, π/2). So invalid.n=1:θ = (4π/3 + 2π*1)/4 = (4π/3 + 6π/3)/4 = (10π/3)/4 = 5π/6. Which is more than π/2. So invalid.Therefore, the only valid solution is θ = π/3. But wait, θ is between 0 and π/2, so π/3 ≈ 1.047 is within that interval. Therefore, θ = π/3.Wait, but let's verify this. Let me plug θ = π/3 into g(x):g(x) = sin(2x - 2θ + π/3) = sin(2x - 2*(π/3) + π/3) = sin(2x - π/3)Now, check the symmetry about (π/6, 0). So, for any x, g(π/3 - x) should equal -g(x).Compute g(π/3 - x) = sin(2*(π/3 - x) - π/3) = sin(2π/3 - 2x - π/3) = sin(π/3 - 2x)On the other hand, -g(x) = -sin(2x - π/3)So we need to check if sin(π/3 - 2x) = -sin(2x - π/3)Note that sin(π/3 - 2x) = sin(- (2x - π/3)) = -sin(2x - π/3). Therefore, yes, this holds. Therefore, θ = π/3 is indeed the solution.Wait, but θ is supposed to be between 0 and π/2. π/3 is approximately 1.047, which is between 0 and π/2 (which is approximately 1.570). So θ = π/3 is valid.Therefore, the answer to part (1) is θ = π/3.But wait, let me make sure. Let's check another x. Suppose x = 0. Then g(0) = sin(-π/3) = -sqrt(3)/2. Then, according to symmetry, g(π/3 - 0) = g(π/3) should be equal to -g(0) = sqrt(3)/2. Let's compute g(π/3):g(π/3) = sin(2*(π/3) - π/3) = sin(2π/3 - π/3) = sin(π/3) = sqrt(3)/2. Which matches. So the symmetry holds here.Another test: x = π/6. Then g(π/6) = sin(2*(π/6) - π/3) = sin(π/3 - π/3) = sin(0) = 0. Then, symmetry implies that g(π/3 - π/6) = g(π/6) = 0, and -g(π/6) = 0, which checks out.Another test: x = π/12. Then g(π/12) = sin(2*(π/12) - π/3) = sin(π/6 - π/3) = sin(-π/6) = -1/2. Then, g(π/3 - π/12) = g(π/4) = sin(2*(π/4) - π/3) = sin(π/2 - π/3) = sin(π/6) = 1/2. Which is -g(π/12) = 1/2. Correct. So symmetry holds. Therefore, θ = π/3 is indeed correct.Therefore, part (1) answer is θ = π/3.Now, moving to part (2). It's a triangle problem.Triangle ABC has sides a, b, c opposite angles A, B, C respectively. The circumradius is R = sqrt(21)/3. Given that g(A/8) = -1/2, b = 1, and we need to find the length of AD, where D is the trisection point of side BC closer to B.First, let's recall that in a triangle, the circumradius R is given by the formula a/(2 sinA) = b/(2 sinB) = c/(2 sinC) = 2R. Wait, actually, the formula is a = 2R sinA, b = 2R sinB, c = 2R sinC. So sides are proportional to the sines of their opposite angles.Given that b = 1, R = sqrt(21)/3, so 1 = 2*(sqrt(21)/3)*sinB => sinB = 3/(2 sqrt(21)) = (3)/(2 sqrt(21)) = rationalize denominator: 3 sqrt(21)/(2*21) = sqrt(21)/14. Wait, let's compute that:sinB = 1 / (2*(sqrt(21)/3)) = 1 / (2 sqrt(21)/3) = 3/(2 sqrt(21)) = multiply numerator and denominator by sqrt(21):3 sqrt(21) / (2*21) = sqrt(21)/14. So sinB = sqrt(21)/14.But sqrt(21)/14 is approximately 0.327, which is a valid sine value. Then angle B is arcsin(sqrt(21)/14). Let's keep that in mind.Now, we also have g(A/8) = -1/2. Recall that g(x) = sin(2x - π/3). Wait, from part (1), we found that θ = π/3, so g(x) = f(x - π/3) = sin(2(x - π/3) + π/3) = sin(2x - 2π/3 + π/3) = sin(2x - π/3). Wait, yes, as we derived earlier. So g(x) = sin(2x - π/3). Therefore, the equation g(A/8) = -1/2 translates to:sin(2*(A/8) - π/3) = -1/2 => sin(A/4 - π/3) = -1/2.We need to solve for angle A. Let's solve this equation:sin(α) = -1/2, where α = A/4 - π/3.The general solution for sinα = -1/2 is α = 7π/6 + 2πn or α = 11π/6 + 2πn, where n is integer. But since A is an angle in a triangle, it must be between 0 and π. Therefore, α = A/4 - π/3 must satisfy the equation sinα = -1/2, and α must be such that A = 4(α + π/3) is between 0 and π.Let's compute possible α:Case 1: α = 7π/6 + 2πnThen A = 4*(7π/6 + 2πn + π/3) = 4*(7π/6 + π/3 + 2πn) = 4*(7π/6 + 2π/6 + 2πn) = 4*(9π/6 + 2πn) = 4*(3π/2 + 2πn) = 6π + 8πn. But A must be between 0 and π, so no solution here.Case 2: α = 11π/6 + 2πnThen A = 4*(11π/6 + 2πn + π/3) = 4*(11π/6 + 2π/6 + 2πn) = 4*(13π/6 + 2πn) = (26π/3) + 8πn. Again, this is more than 8π/3, which is way beyond π. So n must be negative.Wait, perhaps I made a miscalculation.Wait, original equation:sin(α) = -1/2Solutions: α = 7π/6 + 2πn or α = 11π/6 + 2πnBut α = A/4 - π/3So A/4 - π/3 = 7π/6 + 2πn or A/4 - π/3 = 11π/6 + 2πnSolving for A:A/4 = 7π/6 + π/3 + 2πn = 7π/6 + 2π/6 + 2πn = 9π/6 + 2πn = 3π/2 + 2πnThus, A = 6π + 8πnAlternatively,A/4 = 11π/6 + π/3 + 2πn = 11π/6 + 2π/6 + 2πn = 13π/6 + 2πnThus, A = 26π/3 + 8πnBut A must be between 0 and π. Let's check possible n for the second case:If n = -1:A = 26π/3 + 8π*(-1) = 26π/3 - 8π = 26π/3 - 24π/3 = 2π/3. Which is within (0, π). So A = 2π/3.Similarly, check n = -2:A = 26π/3 + 8π*(-2) = 26π/3 - 16π = 26π/3 - 48π/3 = -22π/3. Negative, invalid.Therefore, the only valid solution is A = 2π/3.Therefore, angle A = 2π/3 radians, which is 120 degrees.So, angle A is 120 degrees.Now, with angle A = 2π/3, angle B we found earlier has sinB = sqrt(21)/14. Let's compute angle B.First, sinB = sqrt(21)/14. Let's compute angle B:We can compute cosB using sin²B + cos²B = 1:cosB = sqrt(1 - (sqrt(21)/14)^2 ) = sqrt(1 - 21/196 ) = sqrt( (196 - 21)/196 ) = sqrt(175/196 ) = sqrt(175)/14 = 5*sqrt(7)/14.Therefore, cosB = 5√7 /14.Since the sum of angles in a triangle is π, angle C = π - A - B = π - 2π/3 - B = π/3 - B.We might need sides a, b, c. Given that b = 1, and R = sqrt(21)/3.Recall that in a triangle, a = 2R sinA, b = 2R sinB, c = 2R sinC.So let's compute the sides.First, a = 2R sinA = 2*(sqrt(21)/3)*sin(2π/3) = 2*(sqrt(21)/3)*(sqrt(3)/2) = (sqrt(21)/3)*sqrt(3) = sqrt(63)/3 = (3*sqrt(7))/3 = sqrt(7). So a = sqrt(7).b is given as 1, which we can verify: b = 2R sinB = 2*(sqrt(21)/3)*(sqrt(21)/14) = 2*(21/42) = 2*(1/2) = 1. Correct.c = 2R sinC. Since angle C = π/3 - B. Therefore, sinC = sin(π/3 - B) = sinπ/3 cosB - cosπ/3 sinB = (sqrt(3)/2)*(5√7/14) - (1/2)*(sqrt(21)/14)Compute term by term:First term: (sqrt(3)/2)*(5√7/14) = (5 sqrt(21))/28Second term: (1/2)*(sqrt(21)/14) = sqrt(21)/28Therefore, sinC = (5 sqrt(21)/28) - (sqrt(21)/28) = (4 sqrt(21))/28 = sqrt(21)/7Therefore, c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/21)/3 = 2*(1)/3 = 2/3. So c = 2/3.Therefore, the sides are:a = sqrt(7), opposite angle A = 120°b = 1, opposite angle B = arcsin(sqrt(21)/14)c = 2/3, opposite angle C = π/3 - BNow, we need to find the length of AD, where D is the trisection point of BC closer to B. So BC is side a, which has length sqrt(7). Wait, hold on:Wait, in triangle ABC, side a is opposite angle A, which is BC. Side b is opposite angle B, which is AC. Side c is opposite angle C, which is AB.So BC = a = sqrt(7), AC = b = 1, AB = c = 2/3.Point D is the trisection point of BC closer to B. So BD = (1/3) BC, DC = (2/3) BC. Since BC = sqrt(7), BD = sqrt(7)/3, DC = 2 sqrt(7)/3.We need to find the length of AD.To find AD, we can use coordinates or vectors or the formula for the length of a median, but adjusted since D is not the midpoint but a trisection point.Alternatively, use the Stewart's theorem. Stewart's theorem relates the lengths of the sides of a triangle to the length of a cevian. In this case, AD is a cevian.Stewart's theorem states that for a triangle with sides a, b, c, and a cevian of length d to the side a, dividing it into lengths m and n, then b²m + c²n = a(d² + mn)In our case, side BC is length a = sqrt(7), divided into BD = m = sqrt(7)/3 and DC = n = 2 sqrt(7)/3. The cevian is AD, which we need to find. The sides adjacent to the cevian are AB = c = 2/3 and AC = b = 1.Wait, but in Stewart's formula, if we consider triangle ABC with side BC = a, then the formula is:AB² * DC + AC² * BD = BC * (AD² + BD * DC)So plugging in the values:AB² * DC + AC² * BD = BC * (AD² + BD * DC)Compute each term:AB = c = 2/3, so AB² = (4/9)DC = 2 sqrt(7)/3AC = b = 1, so AC² = 1BD = sqrt(7)/3BC = sqrt(7)Therefore:Left side: (4/9)*(2 sqrt(7)/3) + (1)*(sqrt(7)/3) = (8 sqrt(7))/27 + sqrt(7)/3 = (8 sqrt(7) + 9 sqrt(7))/27 = 17 sqrt(7)/27Right side: sqrt(7)*(AD² + (sqrt(7)/3)*(2 sqrt(7)/3)) = sqrt(7)*(AD² + (7/9 * 2)) = sqrt(7)*(AD² + 14/9)Therefore:17 sqrt(7)/27 = sqrt(7)*(AD² + 14/9)Divide both sides by sqrt(7):17/27 = AD² + 14/9Solve for AD²:AD² = 17/27 - 14/9 = 17/27 - 42/27 = (17 - 42)/27 = -25/27Wait, that can't be. AD² cannot be negative. This suggests an error in calculation. Let me check step by step.First, Stewart's theorem formula:AB² * DC + AC² * BD = BC * (AD² + BD * DC)Given:AB = 2/3, so AB² = 4/9DC = 2 sqrt(7)/3AC = 1, so AC² = 1BD = sqrt(7)/3BC = sqrt(7)Compute left side:(4/9)*(2 sqrt(7)/3) + (1)*(sqrt(7)/3) = (8 sqrt(7))/27 + sqrt(7)/3Convert sqrt(7)/3 to 9 sqrt(7)/27:So left side = (8 sqrt(7) + 9 sqrt(7))/27 = 17 sqrt(7)/27Right side:sqrt(7)*(AD² + (sqrt(7)/3)*(2 sqrt(7)/3)) = sqrt(7)*(AD² + (7/9 * 2)) ???Wait, (sqrt(7)/3)*(2 sqrt(7)/3) = (2*7)/(9) = 14/9. Therefore:Right side = sqrt(7)*(AD² + 14/9)Therefore, setting equal:17 sqrt(7)/27 = sqrt(7)*(AD² + 14/9)Divide both sides by sqrt(7):17/27 = AD² + 14/9Compute AD²:AD² = 17/27 - 14/9 = 17/27 - 42/27 = -25/27Negative result. Impossible. So there must be a mistake in the setup.Wait, let me verify the sides again.Given:- Angle A = 2π/3 (120°), opposite side a = BC = sqrt(7)- Side b = AC = 1, opposite angle B- Side c = AB = 2/3, opposite angle CTherefore, coordinates:Let me try coordinate geometry.Let me place point B at the origin (0,0), point C at (sqrt(7), 0). Then point D is the trisection point closer to B, so D is located at (sqrt(7)/3, 0).Point A is somewhere in the plane. We need to find coordinates of A, then compute distance from A to D.To find coordinates of A, since we know sides AB = 2/3, AC = 1, and BC = sqrt(7).So coordinates:Let’s denote coordinates:B = (0,0)C = (sqrt(7), 0)D = (sqrt(7)/3, 0)Point A has coordinates (x,y). We need to satisfy distances:From A to B: sqrt(x² + y²) = 2/3From A to C: sqrt( (x - sqrt(7))² + y² ) = 1So we have two equations:1. x² + y² = (2/3)^2 = 4/92. (x - sqrt(7))² + y² = 1^2 = 1Subtract equation 1 from equation 2:(x - sqrt(7))² + y² - x² - y² = 1 - 4/9Expand (x - sqrt(7))² = x² - 2x sqrt(7) + 7Therefore:x² - 2x sqrt(7) + 7 + y² - x² - y² = 5/9Simplify:-2x sqrt(7) + 7 = 5/9Then:-2x sqrt(7) = 5/9 - 7 = 5/9 - 63/9 = -58/9Therefore:x = (-58/9) / (-2 sqrt(7)) = (58/9) / (2 sqrt(7)) = 29/(9 sqrt(7)) = rationalize:29 sqrt(7)/(9*7) = 29 sqrt(7)/63Then, substitute x into equation 1 to find y:x² + y² = 4/9Compute x²:(29 sqrt(7)/63)^2 = (841 * 7)/(63^2) = (5887)/(3969) = approximately 1.484, but let's compute exactly:Wait, 29^2 = 841, 7 stays, denominator is 63^2 = 3969.So x² = 841 * 7 / 3969 = 5887 / 3969. Let's see if this reduces. Let me check:Divide numerator and denominator by 7:5887 ÷ 7 = 841, 3969 ÷7 = 567. So 841/567. Hmm, 841 is 29², 567 is 81 * 7. Not sure if simplifies further.So y² = 4/9 - 5887/3969 = convert 4/9 to 1764/3969.Then y² = (1764 - 5887)/3969 = (-4123)/3969. Negative value, which is impossible.Wait, this can't be. There must be a miscalculation. Let's check the previous steps.From equation 2 - equation 1:(x - sqrt(7))² - x² = 1 - 4/9 => x² - 2x sqrt(7) + 7 - x² = 5/9 => -2x sqrt(7) + 7 = 5/9Hence, -2x sqrt(7) = 5/9 - 7 = 5/9 - 63/9 = -58/9 => x = (-58/9)/(-2 sqrt(7)) = 58/(18 sqrt(7)) = 29/(9 sqrt(7)) = 29 sqrt(7)/63. Correct.Then x = 29 sqrt(7)/63 ≈ 29*2.6458/63 ≈ 76.728/63 ≈ 1.217.But BC is length sqrt(7) ≈ 2.645. So point A has x-coordinate ≈1.217, which is between 0 and sqrt(7). Then, compute y² = 4/9 - x².x² = (29 sqrt(7)/63)^2 = (841 * 7)/ (63^2) = 5887 / 3969 ≈1.484Then y² = 4/9 - 1.484 ≈0.444 -1.484 ≈-1.04. Negative. Which is impossible.This suggests inconsistency in the triangle's side lengths. But we computed sides a, b, c based on the given circumradius R and angle A. How come?Wait, let's re-examine the earlier steps. When we found angle A = 2π/3, then computed sides a, b, c using the sine formula. Given that R = sqrt(21)/3, and angles A, B, C.Wait, in the sine formula, a = 2R sinA = 2*(sqrt(21)/3)*sin(2π/3) = 2*(sqrt(21)/3)*(sqrt(3)/2) = sqrt(21)*sqrt(3)/3 = sqrt(63)/3 = 3*sqrt(7)/3 = sqrt(7). So a = sqrt(7) is correct.Similarly, b = 2R sinB = 2*(sqrt(21)/3)*(sqrt(21)/14) = 2*(21/42) = 1. Correct.c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/21) = 2/3. Correct.So sides are a = sqrt(7), b =1, c=2/3. However, when trying to place the triangle in coordinates, we get an inconsistency. Which suggests that either my coordinate setup is wrong or there's a mistake in the application.Wait, in a triangle, the sum of any two sides must be greater than the third side. Check:a = sqrt(7) ≈2.645, b=1, c=0.666. So a + b > c: 2.645 +1 >0.666, yes.a + c > b: 2.645 +0.666 >1, yes.b + c > a: 1 +0.666 ≈1.666, which is not greater than 2.645. Therefore, this triangle cannot exist. Wait, this is a problem.Wait, according to the triangle inequality, the sum of any two sides must be greater than the third side. Here, b + c = 1 + 2/3 ≈1.666 < a ≈2.645. Therefore, such a triangle cannot exist. This is a contradiction.So where did we go wrong? We computed sides a, b, c using the sine formula, assuming the given data. So perhaps the given data is inconsistent? But the problem says "the triangle ABC has sides a, b, c... the circumradius is R= sqrt(21)/3... g(A/8)= -1/2, b=1, R= sqrt(21)/3". Therefore, according to the problem statement, such a triangle exists. But according to our calculations, the sides violate the triangle inequality.This suggests a miscalculation in our previous steps.Wait, let's double-check the computation of side c.We had angle C = π/3 - B.SinC = sin(π/3 - B) = sinπ/3 cosB - cosπ/3 sinBWe computed sinπ/3 = sqrt(3)/2, cosB = 5 sqrt(7)/14, cosπ/3 = 1/2, sinB = sqrt(21)/14Therefore:sinC = sqrt(3)/2 * 5 sqrt(7)/14 - 1/2 * sqrt(21)/14= (5 sqrt(21))/28 - sqrt(21)/28= 4 sqrt(21)/28 = sqrt(21)/7Therefore, sinC = sqrt(21)/7. Then c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21)/21 = 2/3. Correct.Thus, sides a = sqrt(7), b=1, c=2/3. But sum of b and c is 1 + 2/3 = 5/3 ≈1.666 < sqrt(7) ≈2.645. Violates triangle inequality. Contradiction. Therefore, the problem as stated has inconsistent data. But since the problem says "Let triangle ABC...", implying it exists, there must be an error in my calculations.Wait, perhaps angle A is not 2π/3? Let's revisit the step where we found angle A.We had g(A/8) = -1/2, where g(x) = sin(2x - π/3). Therefore:sin(2*(A/8) - π/3) = sin(A/4 - π/3) = -1/2.We solved this and found A = 2π/3. But let's check if there's another solution for angle A within 0 to π.The general solution for sin(α) = -1/2 is α = 7π/6 + 2πn or α = 11π/6 + 2πn. Then α = A/4 - π/3.So A/4 - π/3 = 7π/6 + 2πn => A/4 = 7π/6 + π/3 + 2πn = 7π/6 + 2π/6 + 2πn = 9π/6 + 2πn = 3π/2 + 2πn => A = 6π + 8πn.But A must be in (0, π). For n=0, A=6π, which is too big. For negative n, n=-1: A=6π -8π= -2π, invalid.Alternative solution:A/4 - π/3 = 11π/6 + 2πn => A/4 = 11π/6 + π/3 + 2πn = 11π/6 + 2π/6 + 2πn = 13π/6 + 2πn => A = 26π/3 + 8πn.For n= -1: A=26π/3 -8π=26π/3 -24π/3=2π/3. Which is valid. For n= -2: A=26π/3 -16π= negative. So only valid solution is A=2π/3.Therefore, angle A must be 2π/3, but this leads to triangle sides violating triangle inequality. Therefore, either there's a mistake in the problem, or in my previous steps.Alternatively, maybe I misapplied Stewart's theorem.Wait, Stewart's theorem formula is:AB² * DC + AC² * BD = BC * (AD² + BD * DC)But BD is the segment adjacent to AB, and DC is adjacent to AC. In our case, BD = sqrt(7)/3, DC = 2 sqrt(7)/3, BC = sqrt(7).AB = 2/3, AC=1.So substituting:( (2/3)^2 ) * (2 sqrt(7)/3 ) + (1)^2 * (sqrt(7)/3 ) = sqrt(7) * (AD² + (sqrt(7)/3)(2 sqrt(7)/3))Compute left side:(4/9)(2 sqrt(7)/3) + (1)(sqrt(7)/3) = (8 sqrt(7))/27 + sqrt(7)/3 = (8 sqrt(7) + 9 sqrt(7))/27 = 17 sqrt(7)/27Right side:sqrt(7) * (AD² + (2*7)/9 ) = sqrt(7) * (AD² + 14/9 )Therefore:17 sqrt(7)/27 = sqrt(7) (AD² + 14/9 )Divide both sides by sqrt(7):17/27 = AD² + 14/9 => AD² = 17/27 - 14/9 = 17/27 - 42/27 = -25/27.Negative value. Therefore, impossible.This suggests that with the given data, the triangle doesn't exist. But the problem states it does. Therefore, likely a mistake in my calculations. Let me check the sides again.Given:- Angle A = 2π/3, circumradius R = sqrt(21)/3.- By the Law of Sines, a = 2R sinA = 2*(sqrt(21)/3)*(sin(2π/3)) = 2*(sqrt(21)/3)*(sqrt(3)/2) = sqrt(21)*sqrt(3)/3 = sqrt(63)/3 = 3*sqrt(7)/3 = sqrt(7). Correct.- b = 2R sinB = 1. Therefore, sinB = 1/(2R) = 1/(2*(sqrt(21)/3)) = 3/(2 sqrt(21)) = sqrt(21)/14. Therefore, angle B = arcsin(sqrt(21)/14). Correct.- Angle C = π - A - B = π - 2π/3 - B = π/3 - B. Then, sinC = sin(π/3 - B) = sin(π/3)cosB - cos(π/3)sinB = (sqrt(3)/2)(sqrt(1 - (sqrt(21)/14)^2)) - (1/2)(sqrt(21)/14)Compute cosB:cosB = sqrt(1 - (sqrt(21)/14)^2 ) = sqrt(1 - 21/196) = sqrt(175/196) = sqrt(175)/14 = 5*sqrt(7)/14. Correct.Then sinC = sqrt(3)/2*(5 sqrt(7)/14) - 1/2*(sqrt(21)/14) = (5 sqrt(21))/28 - sqrt(21)/28 = (4 sqrt(21))/28 = sqrt(21)/7. Correct.Thus, c = 2R sinC = 2*(sqrt(21)/3)*(sqrt(21)/7) = 2*(21/21)/3 = 2/3. Correct.So sides a= sqrt(7), b=1, c=2/3. But b + c = 1 + 2/3 = 5/3 ≈1.666 < a ≈2.645. Violates triangle inequality. Therefore, the triangle is impossible. So the problem is contradictory.But the problem states "Let triangle ABC...", which implies existence. Therefore, there must be a mistake in my solution.Wait, perhaps angle A is not 120 degrees? But we derived angle A = 2π/3 from the given g(A/8) = -1/2. Let's check that again.Given g(x) = sin(2x - π/3). So g(A/8) = sin(2*(A/8) - π/3) = sin(A/4 - π/3) = -1/2.Solutions for sinθ = -1/2 are θ = 7π/6 + 2πn or θ = 11π/6 + 2πn.So θ = A/4 - π/3 = 7π/6 + 2πn or 11π/6 + 2πn.Thus:Case 1: A/4 - π/3 = 7π/6 + 2πn => A/4 = 7π/6 + π/3 + 2πn = 7π/6 + 2π/6 + 2πn = 9π/6 + 2πn = 3π/2 + 2πn => A = 6π + 8πn. Not possible within 0 < A < π.Case 2: A/4 - π/3 = 11π/6 + 2πn => A/4 = 11π/6 + π/3 + 2πn = 11π/6 + 2π/6 + 2πn = 13π/6 + 2πn => A = 26π/3 + 8πn. For n = -1, A = 26π/3 - 8π = 26π/3 - 24π/3 = 2π/3. Which is valid. So angle A = 2π/3 is correct.Therefore, given that angle A = 120°, sides a= sqrt(7), b=1, c=2/3, which fails triangle inequality. Hence, contradiction. This suggests that the problem might have a typo or mistake. However, since the problem is given as such, perhaps I need to proceed despite the inconsistency.Alternatively, perhaps the mistake lies in my interpretation of the problem. Let me re-read the problem."Given f(x)=sin x cos x+sqrt{3}cos^2x - sqrt{3}/2, g(x)=f(x - theta), 0 < theta < pi/2, and the graph of g(x) is symmetric about the point (pi/6,0).(1) Find theta.(2) Let triangle ABC have sides a, b, c opposite angles A, B, C, respectively. The circumradius is R, and g(A/8) = -1/2, b=1, R= sqrt{21}/3. If point D is the trisection point of side BC closer to B, find the length of AD."Wait, maybe in part (2), the function g is different? Wait, no, in part (2), it says "g(A/8) = -1/2". Given that in part (1), we found theta = pi/3, so g(x) is defined as f(x - theta), which is f(x - pi/3) = sin(2x - pi/3). So unless there's a different theta in part (2), but the problem doesn't state that. It seems theta in part (2) is the same theta found in part (1). So in part (2), g(x) is the same function as in part (1), with theta = pi/3. Therefore, my calculation should be correct.But the problem states a triangle with those parameters, which leads to an impossible triangle. This is perplexing.Alternatively, maybe the Law of Sines was applied incorrectly. Let me check.Law of Sines states a/sinA = b/sinB = c/sinC = 2R.Given R = sqrt(21)/3, so 2R = 2*sqrt(21)/3.Therefore:a = 2R sinA = 2*sqrt(21)/3 sinAb = 2R sinB = 1c = 2R sinC = 2*sqrt(21)/3 sinCWe computed sinA = sin(2pi/3) = sqrt(3)/2, so a = 2*sqrt(21)/3*(sqrt(3)/2) = sqrt(21)*sqrt(3)/3 = sqrt(63)/3 = sqrt(7). Correct.Given b = 1 = 2R sinB => sinB = 1/(2R) = 3/(2 sqrt(21)) = sqrt(21)/14. Correct.Then angle B is arcsin(sqrt(21)/14), which is acute. Then angle C = pi - A - B = pi/3 - B. Then sinC = sin(pi/3 - B) = sin(pi/3)cosB - cos(pi/3)sinB = sqrt(3)/2 * 5 sqrt(7)/14 - 1/2 * sqrt(21)/14 = (5 sqrt(21))/28 - sqrt(21)/28 = 4 sqrt(21)/28 = sqrt(21)/7. Therefore, sinC = sqrt(21)/7, so c = 2R sinC = 2*sqrt(21)/3*(sqrt(21)/7) = 2*21/(21) = 2/3. Correct.But then sides a= sqrt(7), b=1, c=2/3. But these sides do not satisfy the triangle inequality. Therefore, such a triangle cannot exist. This suggests that there is a mistake in the problem's given data.Alternatively, perhaps I misinterpreted the function g(x). In part (1), the function g(x) is defined as f(x - theta), which we found to be sin(2x - pi/3). In part (2), they mention "g(A/8) = -1/2". If theta in part (2) is different, but the problem doesn't specify that. It's likely that theta is the same as found in part (1). Therefore, unless there's a different definition, which isn't indicated, the contradiction remains.Alternatively, perhaps the graph being symmetric about (pi/6, 0) in part (1) leads to a different function g(x), but no, we followed through the steps correctly.Alternatively, maybe the given R is different? The problem states R = sqrt(21)/3. If R were different, but no, it's given.Alternatively, perhaps angle A is different. But according to the given function g(x) = sin(2x - pi/3), setting g(A/8) = -1/2 leads uniquely to A = 2pi/3. There's no other solution within 0 < A < pi.Therefore, conclusion: The problem as stated contains conflicting information, making the triangle impossible. However, since the problem asks to find AD, perhaps we proceed formally despite the inconsistency.Alternatively, maybe I made a mistake in coordinates. Let me try using vectors or coordinate geometry.Let me place point B at (0,0), point C at (sqrt(7), 0). Point D is at (sqrt(7)/3, 0). Point A is somewhere in the plane. We have AB = 2/3, AC = 1. This forms two circles:- Circle centered at B with radius 2/3: x² + y² = (2/3)^2 = 4/9.- Circle centered at C with radius 1: (x - sqrt(7))² + y² = 1.Find their intersection point A.Subtracting the two equations:(x - sqrt(7))² + y² - x² - y² = 1 - 4/9Expand (x - sqrt(7))² - x² = -2x sqrt(7) + 7 = 5/9.So -2x sqrt(7) = 5/9 - 7 = -58/9 => x = 58/(18 sqrt(7)) = 29/(9 sqrt(7)) = 29 sqrt(7)/63.Then y² = 4/9 - x² = 4/9 - (841 * 7)/(63^2) = 4/9 - 5887/3969 = (4*441 - 5887)/3969 = (1764 - 5887)/3969 = -4123/3969.Negative y², impossible. Therefore, no solution. Hence, the triangle does not exist.But since the problem gives this triangle, there must be an error. Perhaps in part (2), g(x) is a different function? The problem says "g(x) = f(x - theta)", but theta was found in part (1). But maybe in part (2), theta is different? The problem states part (2) starts with "Let triangle ABC...", and mentions g(A/8) = -1/2. Unless there's a different theta in part (2), which is not specified. However, the problem mentions "g(x) = f(x - theta)", but in part (1), theta is found. Unless in part (2), theta is a different variable, but the problem didn't mention redefining theta. This is unclear.Alternatively, maybe the graph of g(x) being symmetric about (pi/6, 0) implies that g(pi/6 + h) = -g(pi/6 - h). Which is another way to express symmetry about a point. Which might imply that the function has a point of symmetry there, meaning that it's odd function shifted to be centered at pi/6. So perhaps the function g(x) has a period such that it's symmetric about pi/6. But we already used that to find theta.Alternatively, maybe I misapplied Stewart's theorem. Alternatively, use coordinates with a different placement.Alternatively, use vector approach. Let's denote vectors.Let me place point B at the origin, vector B = (0,0). Point C at (sqrt(7),0). Point D is at (sqrt(7)/3, 0). Point A is unknown, with AB = 2/3, AC =1.Let vector A = (x,y). Then:|A - B| = sqrt(x² + y²) = 2/3|A - C| = sqrt( (x - sqrt(7))² + y² ) =1These equations lead to the same inconsistency as before. Thus, the problem is impossible.Given that, but the problem still asks to find AD, perhaps the answer is AD = 2/3. But that's the length of AB. Or perhaps AD is another length. But with the given data, the triangle doesn't exist, so the problem is invalid. However, since the problem is presented as solvable, perhaps there is a mistake in my previous steps.Wait, going back to part (1), maybe theta is not pi/3. Let's re-examine part (1) for possible errors.In part (1), we had f(x) = sinx cosx + sqrt(3) cos²x - sqrt(3)/2. Simplified to sin(2x + pi/3). Therefore, f(x) = sin(2x + pi/3). Then, g(x) = f(x - theta) = sin(2(x - theta) + pi/3) = sin(2x - 2 theta + pi/3). Then, the symmetry about (pi/6, 0) requires that g(pi/3 - x) = -g(x). Therefore:sin(2*(pi/3 - x) - 2 theta + pi/3) = -sin(2x - 2 theta + pi/3)Simplify left side:sin(2pi/3 - 2x - 2 theta + pi/3) = sin(pi - 2x - 2 theta)Right side: -sin(2x - 2 theta + pi/3)So, sin(pi - 2x - 2 theta) = -sin(2x - 2 theta + pi/3)Using identity sin(pi - a) = sin a, so left side: sin(2x + 2 theta)Therefore:sin(2x + 2 theta) = -sin(2x - 2 theta + pi/3)This must hold for all x. Let's set phi = 2x, then equation becomes:sin(phi + 2 theta) = -sin(phi - 2 theta + pi/3)Expand both sides using sine addition formula:Left side: sin(phi + 2 theta) = sin phi cos 2 theta + cos phi sin 2 thetaRight side: -sin(phi - 2 theta + pi/3) = -[ sin(phi - 2 theta) cos(pi/3) + cos(phi - 2 theta) sin(pi/3) ]= -[ sin(phi - 2 theta)*(1/2) + cos(phi - 2 theta)*(sqrt(3)/2) ]= -1/2 sin(phi - 2 theta) - sqrt(3)/2 cos(phi - 2 theta)Thus, equation:sin phi cos 2 theta + cos phi sin 2 theta = -1/2 sin(phi - 2 theta) - sqrt(3)/2 cos(phi - 2 theta)Expand sin(phi - 2 theta) and cos(phi - 2 theta):sin(phi - 2 theta) = sin phi cos 2 theta - cos phi sin 2 thetacos(phi - 2 theta) = cos phi cos 2 theta + sin phi sin 2 thetaTherefore, substitute back:Left side: sin phi cos 2 theta + cos phi sin 2 thetaRight side: -1/2 [ sin phi cos 2 theta - cos phi sin 2 theta ] - sqrt(3)/2 [ cos phi cos 2 theta + sin phi sin 2 theta ]= -1/2 sin phi cos 2 theta + 1/2 cos phi sin 2 theta - sqrt(3)/2 cos phi cos 2 theta - sqrt(3)/2 sin phi sin 2 thetaThus, equation becomes:sin phi cos 2 theta + cos phi sin 2 theta = -1/2 sin phi cos 2 theta + 1/2 cos phi sin 2 theta - sqrt(3)/2 cos phi cos 2 theta - sqrt(3)/2 sin phi sin 2 thetaBring all terms to left side:sin phi cos 2 theta + cos phi sin 2 theta + 1/2 sin phi cos 2 theta - 1/2 cos phi sin 2 theta + sqrt(3)/2 cos phi cos 2 theta + sqrt(3)/2 sin phi sin 2 theta =0Factor terms:sin phi cos 2 theta (1 + 1/2) + cos phi sin 2 theta (1 - 1/2) + sqrt(3)/2 cos phi cos 2 theta + sqrt(3)/2 sin phi sin 2 theta =0Simplify coefficients:(3/2) sin phi cos 2 theta + (1/2) cos phi sin 2 theta + (sqrt(3)/2) cos phi cos 2 theta + (sqrt(3)/2) sin phi sin 2 theta =0Group terms with sin phi and cos phi:sin phi [ (3/2) cos 2 theta + (sqrt(3)/2) sin 2 theta ] + cos phi [ (1/2) sin 2 theta + (sqrt(3)/2) cos 2 theta ] =0This equation must hold for all phi, which is only possible if the coefficients of sin phi and cos phi are both zero. Therefore:(3/2) cos 2 theta + (sqrt(3)/2) sin 2 theta =0and(1/2) sin 2 theta + (sqrt(3)/2) cos 2 theta =0These are two equations. Let's write them:1. 3 cos 2 theta + sqrt(3) sin 2 theta =02. sin 2 theta + sqrt(3) cos 2 theta =0Wait, equation 2 is sin 2 theta + sqrt(3) cos 2 theta =0Equation 1: 3 cos 2 theta + sqrt(3) sin 2 theta =0Let me solve equation 2 first:sin 2 theta + sqrt(3) cos 2 theta =0 => tan 2 theta = -sqrt(3)Therefore, 2 theta = -pi/3 + pi*n => theta = -pi/6 + pi*n/2Given theta is in (0, pi/2), possible solution:n=1: theta = -pi/6 + pi/2 = pi/3n=2: theta = -pi/6 + pi = 5pi/6, which is more than pi/2. So only theta = pi/3.Check equation 1:3 cos 2 theta + sqrt(3) sin 2 theta =0With theta=pi/3:cos 2 theta = cos(2pi/3) = -1/2sin 2 theta = sin(2pi/3) = sqrt(3)/2Thus, 3*(-1/2) + sqrt(3)*(sqrt(3)/2) = -3/2 + 3/2 =0. Correct.Therefore, theta=pi/3 is the solution. Hence, part (1) is correct.Therefore, the issue must be in part (2). Given the problem's inconsistency, but assuming that it's intended to have a solution, perhaps there's a different approach.Alternatively, perhaps using the Law of Cosines despite the triangle inequality issue.Given sides a= sqrt(7), b=1, c=2/3.Law of Cosines for angle A:a² = b² + c² - 2bc cosACompute:7 =1 + 4/9 - 2*(1)*(2/3)*cosA7 =13/9 - (4/3) cosA7 -13/9 = -4/3 cosA(63/9 -13/9)= -4/3 cosA =>50/9 = -4/3 cosA => cosA = - (50/9)*(3/4) = -50/12 = -25/6. Which is impossible, since cosA must be between -1 and 1.This again confirms the triangle cannot exist.Given this inconsistency, but the problem being posed, perhaps there is a typo. Alternatively, maybe in part (2), the value of g(A/8)=-1/2 leads to a different angle A.But as per previous steps, angle A must be 2pi/3. Therefore, the problem has no solution. Hence, I'm stuck.Alternatively, maybe the given R is different. Suppose R is different. But the problem states R= sqrt(21)/3.Alternatively, maybe the given b=1 is different. If b is larger, but no. Given the problem statement, it's challenging to proceed.Alternatively, ignore the inconsistency and compute AD formally. Given that, even if the triangle doesn't exist, perhaps the problem expects an answer of sqrt(AD²) = sqrt(-25/27), but that's imaginary. Which is impossible. Therefore, there must be a mistake in the problem's data.Alternatively, perhaps during the computation of angle A, there's a different solution. Let's re-express the equation sin(A/4 - pi/3) = -1/2.Another approach: since A is between 0 and pi, A/4 is between 0 and pi/4. So A/4 - pi/3 is between -pi/3 and pi/4 - pi/3 = -pi/12. Therefore, the angle alpha = A/4 - pi/3 is in the range (-pi/3, -pi/12). In this interval, sine is negative, which matches the equation sin(alpha) = -1/2. The solutions for sin(alpha) = -1/2 in this interval are alpha = -5pi/6 (but -5pi/6 is less than -pi/3) or alpha = -pi/6.Wait, sin(alpha) = -1/2. In the range (-pi/3, -pi/12), the only solution is alpha = -pi/6, since sin(-pi/6) = -1/2.Therefore, A/4 - pi/3 = -pi/6 => A/4 = pi/3 - pi/6 = pi/6 => A = 2pi/3. So angle A = 2pi/3, same as before. Therefore, no alternative solution. Therefore, contradiction.Therefore, the problem has conflicting information. Since this is an exam-style problem, likely an error in the problem statement. However, given the time invested, perhaps the answer is expected to be AD = 2/3. But given the steps, it's impossible.Alternatively, perhaps the trisection point D is closer to C, but the problem states closer to B. Therefore, BD = 1/3 BC, DC = 2/3 BC. As we considered.Alternatively, use coordinates. But since the triangle is impossible, perhaps the answer is boxed{dfrac{sqrt{21}}{3}}? But this is the circumradius. Alternatively, given the negative value of AD² = -25/27, which is impossible. Therefore, I think the problem is flawed.But assuming that the mistake is in my application of Stewart's theorem, let's try using vectors.Let’s denote vector AB = c = 2/3, vector AC = b = 1. Wait, no, vectors are different. Let's use position vectors.But given the inconsistency, I think it's impossible to find a real answer. Given that, perhaps the answer is boxed{dfrac{sqrt{21}}{6}}? Or another fraction.Alternatively, perhaps there is a calculation mistake in the Stewart's theorem step.Let me recheck Stewart's formula:For triangle ABC with BC = a, BD = m, DC = n, AD = d.Then Stewart's theorem states: AB² * n + AC² * m = BC*(d² + m*n)Given:AB = c = 2/3, AC = b =1, BC = a = sqrt(7), BD = m = sqrt(7)/3, DC = n = 2 sqrt(7)/3Thus:Left side: (2/3)^2 * (2 sqrt(7)/3) + 1^2 * (sqrt(7)/3) = (4/9)(2 sqrt(7)/3) + (1)(sqrt(7)/3) = 8 sqrt(7)/27 + sqrt(7)/3 = 8 sqrt(7)/27 + 9 sqrt(7)/27 = 17 sqrt(7)/27Right side: sqrt(7)*(AD² + (sqrt(7)/3)*(2 sqrt(7)/3)) = sqrt(7)*(AD² + 14/9)Thus, 17 sqrt(7)/27 = sqrt(7)*(AD² + 14/9)Divide both sides by sqrt(7):17/27 = AD² + 14/9AD² = 17/27 - 14/9 = 17/27 - 42/27 = -25/27Negative. Hence, impossible. Therefore, the answer doesn't exist.But since the problem requires an answer, I must have missed something. Perhaps the Law of Cosines with the given data.Alternatively, using the coordinates despite the inconsistency. Maybe there is a complex solution.If we proceed formally, AD = sqrt(-25/27) = (5/3) i sqrt(3)/3. But this is imaginary. Not possible.Given this, I must conclude that the problem has a mistake. However, since the problem is from an exam, perhaps there's a typo in the given data. However, since I have to provide an answer, and based on the given steps, even though the triangle inequality is violated, the answer would be AD = 5/3.Wait, how?Wait, wait, perhaps the calculation of AD is done assuming the triangle exists, despite the inconsistency. For example, if we proceed with the sides, even though they don't form a triangle, compute AD using Stewart's theorem, ignoring the inconsistency.But the result is imaginary. Therefore, it's impossible. However, in some contexts, if the problem is in three-dimensional space, but the problem mentions a triangle, which is two-dimensional.Alternatively, maybe the answer is 5/3. Let me check.If I ignore the previous result and consider that in a valid triangle with sides a, b, c, but this triangle is impossible. However, if there's a miscalculation.Wait, in the Law of Sines, if angle A = 120°, then side a is sqrt(7), which is opposite to 120°, and side b =1, which is opposite to angle B with sinB = sqrt(21)/14. Then, the coordinates are impossible. Therefore, the answer is impossible. Therefore, the problem has no solution. Hence, the answer is imaginary. But since the problem is in a real exam, the answer is expected to be a real number, so I must have made a mistake.Wait, perhaps in the application of g(A/8) = -1/2. Let's re-examine that.g(x) = sin(2x - pi/3). So g(A/8) = sin(A/4 - pi/3) = -1/2.So, solutions for A/4 - pi/3 = 7pi/6 + 2pi n or 11pi/6 + 2pi n. As before.But in my previous calculation, I considered only n=-1 for the second case. But perhaps there's another solution with a different n? Let's see.For example, take theta = A/4 - pi/3 = -pi/6 + 2pi n. Then A/4 = pi/3 - pi/6 + 2pi n = pi/6 + 2pi n. Therefore, A = 2pi/3 + 8pi n. For n=0, A=2pi/3. For n=1, A=2pi/3 +8pi. But A must be less than pi. Therefore, only A=2pi/3 is valid.Hence, angle A=2pi/3 is the only solution. Therefore, the problem's data is inconsistent. Hence, impossible to find real AD.Given that, but the problem is from a source that expects an answer, I'll assume that perhaps the answer is AD=1/2 or another simple fraction, but this is a shot in the dark. Alternatively, maybe the answer is sqrt(7)/3, but that's BD. Alternatively, AD=sqrt( (2/3)^2 + (sqrt(7)/3)^2 - 2*(2/3)*(sqrt(7)/3)*cos(theta) ), but without angle theta, I can't compute.Alternatively, use the Law of Cosines in triangle ABD or ADC. But without knowing angles, it's impossible.Given the time I've spent and the dead end, I'll conclude that there's a mistake in the problem's data, leading to an impossible triangle, and therefore AD cannot be determined. But since the problem asks to find it, perhaps the answer is boxed{dfrac{sqrt{21}}{6}}.But I need to check.Wait, in the impossible triangle, if we proceed formally, AD² = -25/27, which is negative. So AD = 5/(3*sqrt(3)) i. But this is imaginary. Therefore, no real solution.Alternatively, maybe the answer is boxed{dfrac{sqrt{21}}{6}}? Let me see.If I compute BD = sqrt(7)/3, AB = 2/3. Then in triangle ABD, using Law of Cosines:AD² = AB² + BD² - 2 AB BD cos(angle ABD)But angle ABD is angle at B between AB and BD. Which is angle ABC. But angle ABC is angle B, which we have sinB = sqrt(21)/14, cosB = 5 sqrt(7)/14.Therefore, AD² = (2/3)^2 + (sqrt(7)/3)^2 - 2*(2/3)*(sqrt(7)/3)*cosB= 4/9 + 7/9 - (4 sqrt(7)/9)*(5 sqrt(7)/14)= 11/9 - (4 sqrt(7)*5 sqrt(7))/(9*14)= 11/9 - (20]

🔑: Solution:# Part 1: Finding thetaGiven the function f(x)=sin xcos x+sqrt{3}cos^2x-frac{sqrt{3}}{2}, we can simplify it using trigonometric identities:begin{align*}f(x) &= frac{1}{2}sin 2x + frac{sqrt{3}}{2}(2cos^2x - 1) &= frac{1}{2}sin 2x + frac{sqrt{3}}{2}cos 2x &= sinleft(2x+frac{pi}{3}right).end{align*}The function g(x) is defined as g(x)=f(x-theta), which translates to:begin{align*}g(x) &= sinleft(2(x-theta)+frac{pi}{3}right) &= sinleft(2x-2theta+frac{pi}{3}right).end{align*}Given that the graph of g(x) is symmetric about the point left(frac{pi}{6},0right), we set up the equation for symmetry:begin{align*}frac{pi}{3}-2theta+frac{pi}{3} &= kpi -2theta &= kpi - frac{2pi}{3} theta &= -frac{kpi}{2} + frac{pi}{3}.end{align*}Considering the constraint 0 < theta < frac{pi}{2}, we find that theta = frac{pi}{3}.Therefore, boxed{theta = frac{pi}{3}}.# Part 2: Finding the length of ADFrom part 1, we know that g(x) = sinleft(2x-frac{pi}{3}right). Given gleft(frac{A}{8}right) = -frac{1}{2}, we have:begin{align*}sinleft(frac{A}{4}-frac{pi}{3}right) &= -frac{1}{2} frac{A}{4}-frac{pi}{3} &= 2kpi - frac{pi}{6} text{ or } 2kpi - frac{5pi}{6}.end{align*}Solving these equations for A, we get:begin{align*}A &= 8kpi + frac{2pi}{3} text{ or } 8kpi - 2pi.end{align*}Given A in (0, pi), we find A = frac{2pi}{3}.Using the Law of Sines in triangle ABC:begin{align*}a &= 2Rsin A = 2left(frac{sqrt{21}}{3}right)sinleft(frac{2pi}{3}right) = sqrt{7}.end{align*}Since D trisects BC closer to B, BD = frac{a}{3} = frac{sqrt{7}}{3}.Applying the Law of Cosines to find c:begin{align*}7 &= 1 + c^2 + c c &= 2.end{align*}Finding cos B:begin{align*}cos B &= frac{a^2 + c^2 - b^2}{2ac} = frac{7 + 4 - 1}{4sqrt{7}} = frac{5sqrt{7}}{14}.end{align*}Applying the Law of Cosines in triangle ABD to find AD:begin{align*}AD^2 &= 4 + frac{7}{9} - 2 times 2 times frac{sqrt{7}}{3} times frac{5sqrt{7}}{14} = frac{13}{9} AD &= frac{sqrt{13}}{3}.end{align*}Therefore, the length of AD is boxed{frac{sqrt{13}}{3}}.